progression

1. Mat.
sin. suite

Zenbaki arrunten arabera ordenatutako objektuen multzoa (a1, a2, a3, a4... ). Adibideak:
{1, 4, 9, 16, 25...}
{1, 1, 1, 1, 1...}
{x/1, x2/2, x3/3, x4/4...}
Multzoetan bezala, segida elementuz osatuta dago, gai deritzenak; baina, multzoetan ez bezala, ordenak garrantzia du, eta gai berak hainbat aldiz ager daitezke posizio desberdinetan. Gaiak lege edo arau baten arabera eratzen dira. Enegarren gaia an da, n zenbaki arrunta izanik, eta gai orokor deritzo. Segida finitua da gai-kopuru finitua duena. Adibidez, goiko lehenengo adibidekoa. Gai-kopuru infinitua duenari segida infinitu deritzo. Adibidez, goiko bigarren eta hirugarren adibideak.
Segida batzuetan, enegarren gaiak zuzenean adierazten du sortze-araua. Goiko hiru adibideetan, hauek dira enegarren gaiak: lehenengoan, n2; bigarrenean, (1)n + 1; hirugarrenean, xn/n: beti ere, n 1. Segida, beraz, N multzoan definitutako funtzio bat da, eta honela idazten da: ƒ(n) = an.

1. Mat.
Zenbaki arrunten arabera ordenatutako objektuen multzoa (a1, a2, a3, a4... ). Adibideak:
{1, 4, 9, 16, 25...}
{1, 1, 1, 1, 1...}
{x/1, x2/2, x3/3, x4/4...}
Multzoetan bezala, segida elementuz osatuta dago, gai deritzenak; baina, multzoetan ez bezala, ordenak garrantzia du, eta gai berak hainbat aldiz ager daitezke posizio desberdinetan. Gaiak lege edo arau baten arabera eratzen dira. Enegarren gaia an da, n zenbaki arrunta izanik, eta gai orokor deritzo. Segida finitua da gai-kopuru finitua duena. Adibidez, goiko lehenengo adibidekoa. Gai-kopuru infinitua duenari segida infinitu deritzo. Adibidez, goiko bigarren eta hirugarren adibideak.
Segida batzuetan, enegarren gaiak zuzenean adierazten du sortze-araua. Goiko hiru adibideetan, hauek dira enegarren gaiak: lehenengoan, n2; bigarrenean, (1)n + 1; hirugarrenean, xn/n: beti ere, n 1. Segida, beraz, N multzoan definitutako funtzio bat da, eta honela idazten da: ƒ(n) = an.

Segida Edit

Egilea: Patxi Angulo Martin

SEGIDA

Segidak MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHekaaa@375E@  multzoan

Segidak

Definizioa MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHekaaa@375E@  multzoko segida bat MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyfHukaaa@375A@  multzotik MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHekaaa@375E@  multzora doan aplikazio bat da, non n MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiabgIGiolablwriLcaa@39D1@  zenbaki arruntari a n MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBaaaleaacaWGUbaabeaakiabgIGiolabl2riHcaa@3AF1@  elementua esleitzen zaion.

f : , f ( n ) = MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacQdacqWIvesPcqGHsgIRcqWIDesOcaGGSaGaamOzaiaacIcacaWGUbGaaiykaiabg2da9aaa@414D@  segidaren n. MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaaaa@36E1@ gaia = a n ℝ. MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0JaamyyamaaBaaaleaacaWGUbaabeaakiabgIGiolabl2riHcaa@3BF7@

Segidaren elementuei gai deritze, a n MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBaaaleaacaWGUbaabeaakiabgIGiolabl2riHcaa@3AF1@  elementuari, segidaren gai orokor, eta segida { a n } MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiWaaeaacaWGHbWaaSbaaSqaaiaad6gaaeqaaaGccaGL7bGaayzFaaaaaa@3A2E@ idatziko dugu.

Praktikan, f ( n ) MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacIcacaWGUbGaaiykaaaa@3925@  aplikazioa erabili beharrean, f ( ) = { a n } MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacIcacqWIvesPcaGGPaGaeyypa0ZaaiWaaeaacaWGHbWaaSbaaSqaaiaad6gaaeqaaaGccaGL7bGaayzFaaGaeyOGIWSaeSyhHekaaa@4250@ azpimultzoa erabiltzen da.

Adibidea MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHekaaa@375E@ multzoan segida hauek definituko ditugu:

{ a n } = { 1 } = { 1,1,1, ,1, } MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiWaaeaacaWGHbWaaSbaaSqaaiaad6gaaeqaaaGccaGL7bGaayzFaaGaeyypa0ZaaiWaaeaacaaIXaaacaGL7bGaayzFaaGaeyypa0ZaaiWaaeaacaaIXaGaaiilaiaaigdacaGGSaGaaGymaiaacYcacqWIMaYscaGGSaGaaGymaiaacYcacqWIMaYsaiaawUhacaGL9baaaaa@49F7@ segida konstantea da;

{ b n } = { ( 1 ) n } = { 1,1, 1, , ( 1 ) n , }; MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiWaaeaacaWGIbWaaSbaaSqaaiaad6gaaeqaaaGccaGL7bGaayzFaaGaeyypa0ZaaiWaaeaadaqadaqaaiabgkHiTiaaigdaaiaawIcacaGLPaaadaahaaWcbeqaaiaad6gaaaaakiaawUhacaGL9baacqGH9aqpdaGadaqaaiabgkHiTiaaigdacaGGSaGaaGymaiaacYcacqGHsislcaaIXaGaaiilaiablAciljaacYcacaGGOaGaeyOeI0IaaGymaiaacMcadaahaaWcbeqaaiaad6gaaaGccaGGSaGaeSOjGSeacaGL7bGaayzFaaaaaa@52E2@

{ c n } = { ln n n } = { ln n 1 , ln 2 2 , ln 3 3 , , ln n n , }. MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9Gq=xcvk9pIe9q8qqaq=dir=f0=yqaqVeLsFr0=vr0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaGadaqaaiaadogadaWgaaWcbaGaamOBaaqabaaakiaawUhacaGL9baacqGH9aqpdaGadaqaamaalaaabaGaciiBaiaac6gacaWGUbaabaGaamOBaaaaaiaawUhacaGL9baacqGH9aqpdaGadaqaamaalaaabaGaciiBaiaac6gacaWGUbaabaGaaGymaaaacaGGSaWaaSaaaeaaciGGSbGaaiOBaiaaikdaaeaacaaIYaaaaiaacYcadaWcaaqaaiGacYgacaGGUbGaaG4maaqaaiaaiodaaaGaaiilaiablAciljaacYcadaWcaaqaaiGacYgacaGGUbGaamOBaaqaaiaad6gaaaGaaiilaiablAcilbGaay5Eaiaaw2haaaaa@5BE2@

 

Definizioa { b n } MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiWaaeaacaWGIbWaaSbaaSqaaiaad6gaaeqaaaGccaGL7bGaayzFaaaaaa@3A2F@ segida { a n } MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiWaaeaacaWGHbWaaSbaaSqaaiaad6gaaeqaaaGccaGL7bGaayzFaaaaaa@3A2E@ segidaren azpisegida da { b n } MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiWaaeaacaWGIbWaaSbaaSqaaiaad6gaaeqaaaGccaGL7bGaayzFaaaaaa@3A2F@ segidaren gai guztiak { a n } MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiWaaeaacaWGHbWaaSbaaSqaaiaad6gaaeqaaaGccaGL7bGaayzFaaaaaa@3A2E@ segidaren gaiak badira.

Beste era batean, { b n } { a n } MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiWaaeaacaWGIbWaaSbaaSqaaiaad6gaaeqaaaGccaGL7bGaayzFaaGaeyOGIW8aaiWaaeaacaWGHbWaaSbaaSqaaiaad6gaaeqaaaGccaGL7bGaayzFaaaaaa@406B@ betez gero, { b n } MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiWaaeaacaWGIbWaaSbaaSqaaiaad6gaaeqaaaGccaGL7bGaayzFaaaaaa@3A2F@ segida { a n } MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiWaaeaacaWGHbWaaSbaaSqaaiaad6gaaeqaaaGccaGL7bGaayzFaaaaaa@3A2E@ segidaren azpisegida da.

Adibidea { d n } = { ln 2 n 2 n } MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9Gq=xcvk9pIe9q8qqaq=dir=f0=yqaqVeLsFr0=vr0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaGadaqaaiaadsgadaWgaaWcbaGaamOBaaqabaaakiaawUhacaGL9baacqGH9aqpdaGadaqaamaalaaabaGaciiBaiaac6gacaaIYaGaamOBaaqaaiaaikdacaWGUbaaaaGaay5Eaiaaw2haaaaa@461A@ segida { c n } = { ln n n } MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9Gq=xcvk9pIe9q8qqaq=dir=f0=yqaqVeLsFr0=vr0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaGadaqaaiaadogadaWgaaWcbaGaamOBaaqabaaakiaawUhacaGL9baacqGH9aqpdaGadaqaamaalaaabaGaciiBaiaac6gacaWGUbaabaGaamOBaaaaaiaawUhacaGL9baaaaa@44A1@ segidaren azpisegida da.

c 1 = ln 1 1 ,   c 2 = ln 2 2 ,   c 3 = ln 3 3 ,   c 4 = ln 4 4 ,   c 5 = ln 5 5 ,   c 6 = ln 6 6 ,   c 7 = ln 7 7 ,   c 8 = ln 8 8 , MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9Gq=xcvk9pIe9q8qqaq=dir=f0=yqaqVeLsFr0=vr0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGJbWaaSbaaSqaaiaaigdaaeqaaOGaeyypa0ZaaSaaaeaaciGGSbGaaiOBaiaaigdaaeaacaaIXaaaaiaacYcacaqGGaGaam4yamaaBaaaleaacaaIYaaabeaakiabg2da9maalaaabaGaciiBaiaac6gacaaIYaaabaGaaGOmaaaacaGGSaGaaeiiaiaadogadaWgaaWcbaGaaG4maaqabaGccqGH9aqpdaWcaaqaaiGacYgacaGGUbGaaG4maaqaaiaaiodaaaGaaiilaiaabccacaWGJbWaaSbaaSqaaiaaisdaaeqaaOGaeyypa0ZaaSaaaeaaciGGSbGaaiOBaiaaisdaaeaacaaI0aaaaiaacYcacaqGGaGaam4yamaaBaaaleaacaaI1aaabeaakiabg2da9maalaaabaGaciiBaiaac6gacaaI1aaabaGaaGynaaaacaGGSaGaaeiiaiaadogadaWgaaWcbaGaaGOnaaqabaGccqGH9aqpdaWcaaqaaiGacYgacaGGUbGaaGOnaaqaaiaaiAdaaaGaaiilaiaabccacaWGJbWaaSbaaSqaaiaaiEdaaeqaaOGaeyypa0ZaaSaaaeaaciGGSbGaaiOBaiaaiEdaaeaacaaI3aaaaiaacYcacaqGGaGaam4yamaaBaaaleaacaaI4aaabeaakiabg2da9maalaaabaGaciiBaiaac6gacaaI4aaabaGaaGioaaaacaGGSaGaeS47IWeaaa@77CD@

d 1 = ln 2 2 ,  d 2 = ln 4 4 ,  d 3 = ln 6 6 ,   d 4 = ln 8 8 ,   MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9Gq=xcvk9pIe9q8qqaq=dir=f0=yqaqVeLsFr0=vr0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGKbWaaSbaaSqaaiaaigdaaeqaaOGaeyypa0ZaaSaaaeaaciGGSbGaaiOBaiaaikdaaeaacaaIYaaaaiaacYcacaqGGaGaaeizamaaBaaaleaacaaIYaaabeaakiabg2da9maalaaabaGaciiBaiaac6gacaaI0aaabaGaaGinaaaacaGGSaGaaeiiaiaabsgadaWgaaWcbaGaaG4maaqabaGccqGH9aqpdaWcaaqaaiGacYgacaGGUbGaaGOnaaqaaiaaiAdaaaGaaiilaiaabccacaWGKbWaaSbaaSqaaiaaisdaaeqaaOGaeyypa0ZaaSaaaeaaciGGSbGaaiOBaiaaiIdaaeaacaaI4aaaaiaacYcacaqGGaGaeS47IWeaaa@59D1@

Segiden limiteak

Definizioa { a n } MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiWaaeaacaWGHbWaaSbaaSqaaiaad6gaaeqaaaGccaGL7bGaayzFaaaaaa@3A2E@ segida konbergentea da MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHekaaa@375E@  multzoan l MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiBaiabgIGiolabl2riHcaa@39D3@  badago, non l MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiBaaaa@36DF@-ren edozein ingurune irekitan { a n } MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiWaaeaacaWGHbWaaSbaaSqaaiaad6gaaeqaaaGccaGL7bGaayzFaaaaaa@3A2E@ segidaren gai batetik aurrera segidaren gai guztiak dauden.

Kasu horretan l MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiBaaaa@36DF@ -ri segidaren limite deituko diogu eta

lim n a n = l MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9Gq=xcvk9pIe9q8qqaq=dir=f0=yqaqVeLsFr0=vr0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWfqaqaaiGacYgacaGGPbGaaiyBaaWcbaGaamOBaiabgkziUkabg6HiLcqabaGccaWGHbWaaSbaaSqaaiaad6gaaeqaaOGaeyypa0JaamiBaaaa@44B8@

idatziko dugu. Batzuetan, { a n } l MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiWaaeaacaWGHbWaaSbaaSqaaiaad6gaaeqaaaGccaGL7bGaayzFaaGaeyOKH4QaamiBaaaa@3D0C@ ere adieraziko dugu.

Definiziotik zuzenean adierazpen hau atera dezakegu:

lim n a n = l ε > 0    n 0 ( ε ) / n n 0 ( ε )    | a n l | < ε . MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9Gq=xcvk9pIe9q8qqaq=dir=f0=yqaqVeLsFr0=vr0=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@69CF@

Adibidea Esate baterako, { 2 n + 1 n } MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9Gq=xcvk9pIe9q8qqaq=dir=f0=yqaqVeLsFr0=vr0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaGadaqaamaalaaabaGaaGOmaiaad6gacqGHRaWkcaaIXaaabaGaamOBaaaaaiaawUhacaGL9baaaaa@3FCE@ segidaren limitea 2 da; hau da, lim n 2 n + 1 n = 2. MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9Gq=xcvk9pIe9q8qqaq=dir=f0=yqaqVeLsFr0=vr0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWfqaqaaiGacYgacaGGPbGaaiyBaaWcbaGaamOBaiabgkziUkabg6HiLcqabaGcdaWcaaqaaiaaikdacaWGUbGaey4kaSIaaGymaaqaaiaad6gaaaGaeyypa0JaaGOmaiaac6caaaa@4775@  

Propietateak

  1. { a n } MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiWaaeaacaWGHbWaaSbaaSqaaiaad6gaaeqaaaGccaGL7bGaayzFaaGaeyOGIWSaeSyhHekaaa@3D9A@ segida konbergentea bada, limitea bakarra da.

  2. { a n } MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiWaaeaacaWGHbWaaSbaaSqaaiaad6gaaeqaaaGccaGL7bGaayzFaaGaeyOGIWSaeSyhHekaaa@3D9A@ segida konbergentea bada, bere azpisegida guztiak konbergenteak dira eta segidaren limite bera dute.

  3. { a n } MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiWaaeaacaWGHbWaaSbaaSqaaiaad6gaaeqaaaGccaGL7bGaayzFaaGaeyOGIWSaeSyhHekaaa@3D9A@ segida konbergentea bada, { a n } MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiWaaeaacaWGHbWaaSbaaSqaaiaad6gaaeqaaaGccaGL7bGaayzFaaaaaa@3A2E@ segida bornatua da.

Definizioa { a n } MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiWaaeaacaWGHbWaaSbaaSqaaiaad6gaaeqaaaGccaGL7bGaayzFaaGaeyOGIWSaeSyhHekaaa@3D9A@ segida dibergentea da MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHekaaa@375E@  multzoaren jatorriaren, O-ren, ingurune ireki guztietarako { a n } MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiWaaeaacaWGHbWaaSbaaSqaaiaad6gaaeqaaaGccaGL7bGaayzFaaaaaa@3A2E@ segidaren gai batetik aurrera segidaren gai guztiak kanpoaldean badaude.

Kasu horretan, lim n a n = MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9Gq=xcvk9pIe9q8qqaq=dir=f0=yqaqVeLsFr0=vr0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWfqaqaaiGacYgacaGGPbGaaiyBaaWcbaGaamOBaiabgkziUkabg6HiLcqabaGccaWGHbWaaSbaaSqaaiaad6gaaeqaaOGaeyypa0JaeyOhIukaaa@4538@ idatziko dugu. Batzuetan, { a n } MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiWaaeaacaWGHbWaaSbaaSqaaiaad6gaaeqaaaGccaGL7bGaayzFaaGaeyOKH4QaeyOhIukaaa@3D8C@ ere adieraziko dugu. Kontuan izan behar dugu infinituak bi aukera dituela: + MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaey4kaSIaeyOhIukaaa@3841@  eta ∞. MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0IaeyOhIukaaa@384C@

Definizio horretatik adierazpen hau atera dezakegu:

{ a n } MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiWaaeaacaWGHbWaaSbaaSqaaiaad6gaaeqaaaGccaGL7bGaayzFaaaaaa@3A2E@ dibergentea da K > 0 MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyi1HSTaeyiaIiIaam4saiabg6da+iaaicdaaaa@3BAC@   n 0 ( K ) / n n 0 ( K ) MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaey4aIqIaamOBamaaBaaaleaacaaIWaaabeaakmaabmaabaGaam4saaGaayjkaiaawMcaaiabgIGiolablwriLkaac+cacqGHaiIicaWGUbGaeyyzImRaamOBamaaBaaaleaacaaIWaaabeaakmaabmaabaGaam4saaGaayjkaiaawMcaaaaa@4667@   | a n | > K. MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqWaaeaacaWGHbWaaSbaaSqaaiaad6gaaeqaaaGccaGLhWUaayjcSdGaeyOpa4Jaam4saaaa@3CF7@

Adibidea { ln n } MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiWaaeaaciGGSbGaaiOBaiaad6gaaiaawUhacaGL9baaaaa@3AF6@ segida dibergentea da.

Definizioa { a n } MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiWaaeaacaWGHbWaaSbaaSqaaiaad6gaaeqaaaGccaGL7bGaayzFaaGaeyOGIWSaeSyhHekaaa@3D9A@ segida oszilatzailea da ez bada ez konbergentea, ezta dibergentea ere.

Adibidea { ( 1 ) n } MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiWaaeaadaqadaqaaiabgkHiTiaaigdaaiaawIcacaGLPaaadaahaaWcbeqaaiaad6gaaaaakiaawUhacaGL9baaaaa@3C7A@ segida oszilatzailea da.

Segida monotonoak

MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHekaaa@375E@ multzoan MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyizImkaaa@37A3@ (txikiago edo berdin) ordena-erlazioa dago definiturik. Atal honetan, ordena-erlazioak segiden jokaeran duen eragina aztertuko dugu.

Definizioa Izan bedi { a n } MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiWaaeaacaWGHbWaaSbaaSqaaiaad6gaaeqaaaGccaGL7bGaayzFaaGaeSyhHekaaa@3B9E@  multzoko segida bat,

{ a n } MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiWaaeaacaWGHbWaaSbaaSqaaiaad6gaaeqaaaGccaGL7bGaayzFaaaaaa@3A2E@ segida monotono gorakorra da   n n 0 MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyiaIiIaamOBaiabgwMiZkaad6gadaWgaaWcbaGaaGimaaqabaaaaa@3B50@    a n a n + 1 MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBaaaleaacaWGUbaabeaakiabgsMiJkaadggadaWgaaWcbaGaamOBaiabgUcaRiaaigdaaeqaaaaa@3D54@  betetzen badu.

{ a n } MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiWaaeaacaWGHbWaaSbaaSqaaiaad6gaaeqaaaGccaGL7bGaayzFaaaaaa@3A2E@ segida hertsiki monotono gorakorra da   n n 0 MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyiaIiIaamOBaiabgwMiZkaad6gadaWgaaWcbaGaaGimaaqabaaaaa@3B50@   a n < a n + 1 MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBaaaleaacaWGUbaabeaakiabgYda8iaadggadaWgaaWcbaGaamOBaiabgUcaRiaaigdaaeqaaaaa@3CA3@  betetzen badu.

{ a n } MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiWaaeaacaWGHbWaaSbaaSqaaiaad6gaaeqaaaGccaGL7bGaayzFaaaaaa@3A2E@ segida monotono beherakorra da   n n 0 MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyiaIiIaamOBaiabgwMiZkaad6gadaWgaaWcbaGaaGimaaqabaaaaa@3B50@   a n a n + 1 MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBaaaleaacaWGUbaabeaakiabgwMiZkaadggadaWgaaWcbaGaamOBaiabgUcaRiaaigdaaeqaaaaa@3D65@  betetzen badu.

{ a n } MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiWaaeaacaWGHbWaaSbaaSqaaiaad6gaaeqaaaGccaGL7bGaayzFaaaaaa@3A2E@ segida hertsiki monotono beherakorra da   n n 0 MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyiaIiIaamOBaiabgwMiZkaad6gadaWgaaWcbaGaaGimaaqabaaaaa@3B50@   a n > a n + 1 MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBaaaleaacaWGUbaabeaakiabg6da+iaadggadaWgaaWcbaGaamOBaiabgUcaRiaaigdaaeqaaaaa@3CA7@  betetzen badu.

Adibidea

{ 1 } MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiWaaeaacaaIXaaacaGL7bGaayzFaaaaaa@38DA@ segida konstantea monotono gorakorra eta beherakorra da, definizioen arabera.

{ n 1 n } MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9Gq=xcvk9pIe9q8qqaq=dir=f0=yqaqVeLsFr0=vr0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaGadaqaamaalaaabaGaamOBaiabgkHiTiaaigdaaeaacaWGUbaaaaGaay5Eaiaaw2haaaaa@3F1D@ segida hertsiki monotono gorakorra da.

{ n + 1 n } MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9Gq=xcvk9pIe9q8qqaq=dir=f0=yqaqVeLsFr0=vr0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaGadaqaamaalaaabaGaamOBaiabgUcaRiaaigdaaeaacaWGUbaaaaGaay5Eaiaaw2haaaaa@3F12@ segida hertsiki monotono beherakorra da.

Propietatea Segida monotono bornatu guztiak konbergenteak dira.

Adibidea { ( 1 + 1 n ) n } MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9Gq=xcvk9pIe9q8qqaq=dir=f0=yqaqVeLsFr0=vr0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaGadaqaamaabmaabaGaaGymaiabgUcaRmaalaaabaGaaGymaaqaaiaad6gaaaaacaGLOaGaayzkaaWaaWbaaSqabeaacaWGUbaaaaGccaGL7bGaayzFaaaaaa@418D@ segida konbergentea da.

Alde batetik, segida hertsiki monotono gorakorra da eta, beste aldetik, bornatua da

n      2 ( 1 + 1 n ) n < 3 MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9Gq=xcvk9pIe9q8qqaq=dir=f0=yqaqVeLsFr0=vr0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqGHaiIicaWGUbGaeyicI4SaeSyfHuQaaeiiaiaabccacaqGGaGaaeiiaiaaikdacqGHKjYOdaqadaqaaiaaigdacqGHRaWkdaWcaaqaaiaaigdaaeaacaWGUbaaaaGaayjkaiaawMcaamaaCaaaleqabaGaamOBaaaakiabgYda8iaaiodaaaa@4ACD@

betetzen delako. Hortaz, segida konbergentea da. Bere limitea hau da:

lim n ( 1 + 1 n ) n = e = 2 , 71828182 ... MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9Gq=xcvk9pIe9q8qqaq=dir=f0=yqaqVeLsFr0=vr0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWfqaqaaiGacYgacaGGPbGaaiyBaaWcbaGaamOBaiabgkziUkabg6HiLcqabaGcdaqadaqaaiaaigdacqGHRaWkdaWcaaqaaiaaigdaaeaacaWGUbaaaaGaayjkaiaawMcaamaaCaaaleqabaGaamOBaaaakiabg2da9iaadwgacqGH9aqpcaqGYaGaaiilaiaabEdacaqGXaGaaeioaiaabkdacaqG4aGaaeymaiaabIdacaqGYaGaaiOlaiaac6cacaGGUaaaaa@52EE@

Propietateak

  1. { a n } MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiWaaeaacaWGHbWaaSbaaSqaaiaad6gaaeqaaaGccaGL7bGaayzFaaaaaa@3A2E@ segida monotono konbergentearen limitea zero ez bada, segidaren gai batetik aurrera gai guztiek limitearen zeinu bera dute.

  2. { a n } MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiWaaeaacaWGHbWaaSbaaSqaaiaad6gaaeqaaaGccaGL7bGaayzFaaaaaa@3A2E@ eta { b n } MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiWaaeaacaWGIbWaaSbaaSqaaiaad6gaaeqaaaGccaGL7bGaayzFaaaaaa@3A2F@ segida monotono konbergenteen limiteak a MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabmqaamaabaabaaGcbaGaamyyaaaa@36D6@ eta b MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabmqaamaabaabaaGcbaGaamOyaaaa@36D7@ badira eta { c n } MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiWaaeaacaWGJbWaaSbaaSqaaiaad6gaaeqaaaGccaGL7bGaayzFaaaaaa@3A30@ segida monotonoak n n 0 MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyiaIiIaamOBaiabgwMiZkaad6gadaWgaaWcbaGaaGimaaqabaaaaa@3B50@   a n c n b n MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBaaaleaacaWGUbaabeaakiabgsMiJkaadogadaWgaaWcbaGaamOBaaqabaGccqGHKjYOcaWGIbWaaSbaaSqaaiaad6gaaeqaaaaa@3F7E@ betetzen badu, { c n } MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiWaaeaacaWGJbWaaSbaaSqaaiaad6gaaeqaaaGccaGL7bGaayzFaaaaaa@3A30@ segida konbergentea da eta c bere limiteak hau betetzen du: a c b. MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiabgsMiJkaadogacqGHKjYOcaWGIbaaaa@3C0D@

Segiden arteko eragiketak eta limiteak

Definizioa { a n } MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiWaaeaacaWGHbWaaSbaaSqaaiaad6gaaeqaaaGccaGL7bGaayzFaaaaaa@3A2E@ eta { b n } MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiWaaeaacaWGIbWaaSbaaSqaaiaad6gaaeqaaaGccaGL7bGaayzFaaaaaa@3A2F@ segidak emanik, eragiketa hauek definituko ditugu:

Batuketa/kenketa:         { a n } ± { b n } = { a n ± b n }. MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiWaaeaacaWGHbWaaSbaaSqaaiaad6gaaeqaaaGccaGL7bGaayzFaaGaeyySae7aaiWaaeaacaWGIbWaaSbaaSqaaiaad6gaaeqaaaGccaGL7bGaayzFaaGaeyypa0ZaaiWaaeaacaWGHbWaaSbaaSqaaiaad6gaaeqaaOGaeyySaeRaamOyamaaBaaaleaacaWGUbaabeaaaOGaay5Eaiaaw2haaaaa@49A1@

Biderketa:         { a n } × { b n } = { a n b n }. MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiWaaeaacaWGHbWaaSbaaSqaaiaad6gaaeqaaaGccaGL7bGaayzFaaGaey41aq7aaiWaaeaacaWGIbWaaSbaaSqaaiaad6gaaeqaaaGccaGL7bGaayzFaaGaeyypa0ZaaiWaaeaacaWGHbWaaSbaaSqaaiaad6gaaeqaaOGaeyyXICTaamOyamaaBaaaleaacaWGUbaabeaaaOGaay5Eaiaaw2haaaaa@4A26@

Zatiketa:           { a n } ÷ { b n } = { a n ÷ b n }, MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiWaaeaacaWGHbWaaSbaaSqaaiaad6gaaeqaaaGccaGL7bGaayzFaaGaey49aG7aaiWaaeaacaWGIbWaaSbaaSqaaiaad6gaaeqaaaGccaGL7bGaayzFaaGaeyypa0ZaaiWaaeaacaWGHbWaaSbaaSqaaiaad6gaaeqaaOGaey49aGRaamOyamaaBaaaleaacaWGUbaabeaaaOGaay5Eaiaaw2haaaaa@4A3B@ b n 0 MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyamaaBaaaleaacaWGUbaabeaakiabgcMi5kaaicdaaaa@3A7F@  izanik.

Logaritmoa:       log b { a n } = { log b a n }, MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiBaiaac+gacaGGNbWaaSbaaSqaaiaadkgaaeqaaOWaaiWaaeaacaWGHbWaaSbaaSqaaiaad6gaaeqaaaGccaGL7bGaayzFaaGaeyypa0ZaaiWaaeaaciGGSbGaai4BaiaacEgadaWgaaWcbaGaamOyaaqabaGccaWGHbWaaSbaaSqaaiaad6gaaeqaaaGccaGL7bGaayzFaaaaaa@474E@ b > 0 MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyaiabg6da+iaaicdaaaa@3897@  eta a n > 0 MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBaaaleaacaWGUbaabeaakiabg6da+iaaicdaaaa@39BF@  izanik.

Esponentziala:  K { a n } = { K a n } ,    K > 0 MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9Gq=xcvk9pIe9q8qqaq=dir=f0=yqaqVeLsFr0=vr0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGlbWaaWbaaSqabeaadaGadaqaaiaadggadaWgaaadbaGaamOBaaqabaaaliaawUhacaGL9baaaaGccqGH9aqpdaGadaqaaiaadUeadaahaaWcbeqaaiaadggadaWgaaadbaGaamOBaaqabaaaaaGccaGL7bGaayzFaaGaaiilaiaabccacaqGGaGaam4saiabg6da+iaaicdaaaa@4963@ izanik.

Berreketa:         { a n } { b n } = { a n b n } ,    a n > 0 MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9Gq=xcvk9pIe9q8qqaq=dir=f0=yqaqVeLsFr0=vr0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaGadaqaaiaadggadaWgaaWcbaGaamOBaaqabaaakiaawUhacaGL9baadaahaaWcbeqaamaacmaabaGaamOyamaaBaaameaacaWGUbaabeaaaSGaay5Eaiaaw2haaaaakiabg2da9maacmaabaGaamyyamaaDaaaleaacaWGUbaabaGaamOyamaaBaaameaacaWGUbaabeaaaaaakiaawUhacaGL9baacaGGSaGaaeiiaiaabccacaWGHbWaaSbaaSqaaiaad6gaaeqaaOGaeyOpa4JaaGimaaaa@4F1D@ izanik.

Oro har, “segiden arteko eragiketen limitea = limiteen arteko eragiketa” betetzen da. Baina, salbuespenak daude, eta eragiketa bakoitzak bere berezitasunak ditu. Bestalde, eragiketan limite infinitua duen segida sartzen badugu, limitearen kalkulua zaildu egin daiteke. Eragiketa baten limitea aurreko erregela erabiliz kalkulatu ezin denean, indeterminazio bat sortzen dela esango dugu.

Eragiketen limiteetan sor daitezkeen indeterminazioak hauek dira:

Batuketa/kenketa:         ∞; MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOhIuQaeyOeI0IaeyOhIukaaa@39BD@

Biderketa:         0 ∞; MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimaiabgwSixlabg6HiLcaa@3A63@

Zatiketa:           0 0 ,   0 ∞; MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaaIWaaabaGaaGimaaaacaGGSaGaaeiiamaalaaabaGaaGimaaqaaiabg6HiLcaaaaa@3B00@

Berreketa:         0 0 ,   1 ,   0. MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimamaaCaaaleqabaGaaGimaaaakiaacYcacaqGGaGaaGymamaaCaaaleqabaGaeyOhIukaaOGaaiilaiaabccacqGHEisPdaahaaWcbeqaaiaaicdaaaaaaa@3EFA@

Indeterminazioak ebazteko metodoak

Atal honetan, zenbait metodo aztertuko ditugu indeterminazioak ebazteko.

Baliokidetasuna

Definizioa { a n } MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiWaaeaacaWGHbWaaSbaaSqaaiaad6gaaeqaaaGccaGL7bGaayzFaaaaaa@3A2E@ eta { b n } MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiWaaeaacaWGIbWaaSbaaSqaaiaad6gaaeqaaaGccaGL7bGaayzFaaaaaa@3A2F@ segidak baliokideak dira lim n a n b n = 1 MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9Gq=xcvk9pIe9q8qqaq=dir=f0=yqaqVeLsFr0=vr0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWfqaqaaiGacYgacaGGPbGaaiyBaaWcbaGaamOBaiabgkziUkabg6HiLcqabaGcdaWcaaqaaiaadggadaWgaaWcbaGaamOBaaqabaaakeaacaWGIbWaaSbaaSqaaiaad6gaaeqaaaaakiabg2da9iaaigdaaaa@46A2@ denean.

Honela adieraziko dugu:

{ a n } { b n } lim n a n b n = 1 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9Gq=xcvk9pIe9q8qqaq=dir=f0=yqaqVeLsFr0=vr0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaGadaqaaiaadggadaWgaaWcbaGaamOBaaqabaaakiaawUhacaGL9baacqWI8iIodaGadaqaaiaadkgadaWgaaWcbaGaamOBaaqabaaakiaawUhacaGL9baacqGHqhc3cqGHshI3daWfqaqaaiGacYgacaGGPbGaaiyBaaWcbaGaamOBaiabgkziUkabg6HiLcqabaGcdaWcaaqaaiaadggadaWgaaWcbaGaamOBaaqabaaakeaacaWGIbWaaSbaaSqaaiaad6gaaeqaaaaakiabg2da9iaaigdaaaa@5501@

Propietatea { a n } MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiWaaeaacaWGHbWaaSbaaSqaaiaad6gaaeqaaaGccaGL7bGaayzFaaaaaa@3A2E@ eta { b n } MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiWaaeaacaWGIbWaaSbaaSqaaiaad6gaaeqaaaGccaGL7bGaayzFaaaaaa@3A2F@ segida baliokideak badira, limite berdina dute.

Adibideak

  1. { 1 n } MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9Gq=xcvk9pIe9q8qqaq=dir=f0=yqaqVeLsFr0=vr0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaGadaqaamaalaaabaGaaGymaaqaaiaad6gaaaaacaGL7bGaayzFaaaaaa@3D3C@ eta { sin 1 n } MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9Gq=xcvk9pIe9q8qqaq=dir=f0=yqaqVeLsFr0=vr0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaGadaqaaiGacohacaGGPbGaaiOBamaalaaabaGaaGymaaqaaiaad6gaaaaacaGL7bGaayzFaaaaaa@4014@ segidak baliokideak dira.

  2. { 1 n } MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9Gq=xcvk9pIe9q8qqaq=dir=f0=yqaqVeLsFr0=vr0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaGadaqaamaalaaabaGaaGymaaqaaiaad6gaaaaacaGL7bGaayzFaaaaaa@3D3C@ eta { 1 n 2 } MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiWaaeaadaWcaaqaaiaaigdaaeaacaWGUbWaaWbaaSqabeaacaaIYaaaaaaaaOGaay5Eaiaaw2haaaaa@3ACF@ ez dira segida baliokideak.

Ordezkapen-printzipioa Segida baten limitea kalkulatzean, gai orokorraren adierazpenean agertzen den biderkagai edo zatitzaile bat bere baliokide batez ordezka daiteke segidaren limitea aldatu gabe.

Definizioa

  1. { a n } MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiWaaeaacaWGHbWaaSbaaSqaaiaad6gaaeqaaaGccaGL7bGaayzFaaaaaa@3A2E@ segida infinitesimala da lim n a n = 0 MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9Gq=xcvk9pIe9q8qqaq=dir=f0=yqaqVeLsFr0=vr0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWfqaqaaiGacYgacaGGPbGaaiyBaaWcbaGaamOBaiabgkziUkabg6HiLcqabaGccaWGHbWaaSbaaSqaaiaad6gaaeqaaOGaeyypa0JaaGimaaaa@4481@  bada.

  2. { a n } MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiWaaeaacaWGHbWaaSbaaSqaaiaad6gaaeqaaaGccaGL7bGaayzFaaaaaa@3A2E@ segida infinitua da lim n a n = MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9Gq=xcvk9pIe9q8qqaq=dir=f0=yqaqVeLsFr0=vr0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWfqaqaaiGacYgacaGGPbGaaiyBaaWcbaGaamOBaiabgkziUkabg6HiLcqabaGccaWGHbWaaSbaaSqaaiaad6gaaeqaaOGaeyypa0JaeyOhIukaaa@4538@  bada ( + MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeqaaeaacqGHRaWkcqGHEisPaiaawIcaaaaa@3907@  edo ). MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeGaaeaacqGHsislcqGHEisPaiaawMcaaaaa@3914@

Segida infinitesimal hauek baliokideak dira:

  1. { a n } { sin a n } { tan  a n } { arcsin  a n } { arctan  a n }. MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiWaaeaacaWGHbWaaSbaaSqaaiaad6gaaeqaaaGccaGL7bGaayzFaaGaeSipIOZaaiWaaeaaciGGZbGaaiyAaiaac6gacaWGHbWaaSbaaSqaaiaad6gaaeqaaaGccaGL7bGaayzFaaGaeSipIOZaaiWaaeaaciGG0bGaaiyyaiaac6gacaWGHbWaaSbaaSqaaiaad6gaaeqaaaGccaGL7bGaayzFaaGaeSipIOZaaiWaaeaaciGGHbGaaiOCaiaacogacaGGZbGaaiyAaiaac6gacaWGHbWaaSbaaSqaaiaad6gaaeqaaaGccaGL7bGaayzFaaGaeSipIOZaaiWaaeaaciGGHbGaaiOCaiaacogacaGG0bGaaiyyaiaac6gacaWGHbWaaSbaaSqaaiaad6gaaeqaaaGccaGL7bGaayzFaaaaaa@60A8@

  2. { 1 cos  a n } { a n 2 2 }. MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiWaaeaacaaIXaGaeyOeI0Iaci4yaiaac+gacaGGZbGaamyyamaaBaaaleaacaWGUbaabeaaaOGaay5Eaiaaw2haaiablYJi6maacmaabaWaaSaaaeaacaWGHbWaa0baaSqaaiaad6gaaeaacaaIYaaaaaGcbaGaaGOmaaaaaiaawUhacaGL9baaaaa@459B@

  3. { a n } { ln ( 1 + a n ) }. MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiWaaeaacaWGHbWaaSbaaSqaaiaad6gaaeqaaaGccaGL7bGaayzFaaGaeSipIOZaaiWaaeaaciGGSbGaaiOBamaabmaabaGaaGymaiabgUcaRiaadggadaWgaaWcbaGaamOBaaqabaaakiaawIcacaGLPaaaaiaawUhacaGL9baaaaa@44A1@

  4. { b n 1 } { ln b n } ,   { b n } 1 MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiWaaeaacaWGIbWaaSbaaSqaaiaad6gaaeqaaOGaeyOeI0IaaGymaaGaay5Eaiaaw2haaiablYJi6maacmaabaGaciiBaiaac6gacaWGIbWaaSbaaSqaaiaad6gaaeqaaaGccaGL7bGaayzFaaGaaiilaiaabccadaGadaqaaiaadkgadaWgaaWcbaGaamOBaaqabaaakiaawUhacaGL9baacqGHsgIRcaaIXaaaaa@4B61@ denean.

Segida infinitu hauek baliokideak dira:

  1. { a k n k + a k 1 n k 1 + a 1 n + a 0 } { a k n k }. MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiWaaeaacaWGHbWaaSbaaSqaaiaadUgaaeqaaOGaamOBamaaCaaaleqabaGaam4AaaaakiabgUcaRiaadggadaWgaaWcbaGaam4AaiabgkHiTiaaigdaaeqaaOGaamOBamaaCaaaleqabaGaam4AaiabgkHiTiaaigdaaaGccqGHRaWkcqWIVlctcaWGHbWaaSbaaSqaaiaaigdaaeqaaOGaamOBaiabgUcaRiaadggadaWgaaWcbaGaaGimaaqabaaakiaawUhacaGL9baacqWI8iIodaGadaqaaiaadggadaWgaaWcbaGaam4AaaqabaGccaWGUbWaaWbaaSqabeaacaWGRbaaaaGccaGL7bGaayzFaaaaaa@546F@

  2. Stirlingen baliokidetza: { n ! } { n n e n 2 π n }. MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9Gq=xcvk9pIe9q8qqaq=dir=f0=yqaqVeLsFr0=vr0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaGadaqaaiaad6gacaGGHaaacaGL7bGaayzFaaGaeSipIOZaaiWaaeaacaWGUbWaaWbaaSqabeaacaWGUbaaaOGaamyzamaaCaaaleqabaGaeyOeI0IaamOBaaaakmaakaaabaGaaGOmaiabec8aWjaad6gaaSqabaaakiaawUhacaGL9baaaaa@4920@

Polinomioen arteko zatiduraren limitea

lim n a k n k + a k 1 n k 1 + a 1 n + a 0 b l n l + b l 1 n l 1 + + b 1 n + b 0 = { k > l  denean a k b k k = l  denean 0 k < l  denean MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9Gq=xcvk9pIe9q8qqaq=dir=f0=yqaqVeLsFr0=vr0=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@8EE7@

Infinituen ordenak

Beste aldetik, infinituak ez dira “abiadura” berdinean hurbiltzen infinitura; “hurbiltze-abiadura” horri infinituaren ordena deituko diogu. Lau ordena bereiziko ditugu:

{ ( ln a n ) q } < < < { ( a n ) p } < < < { k a n } < < < { ( a n ) r a n }, MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9Gq=xcvk9pIe9q8qqaq=dir=f0=yqaqVeLsFr0=vr0=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@609D@

q , p , r > 0 MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyCaiaacYcacaWGWbGaaiilaiaadkhacqGH+aGpcaaIWaaaaa@3BF2@ eta   k > 1 MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Aaiabg6da+iaaigdaaaa@38A1@ izanik.

Stolzen irizpideak

Teorema { a n } MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiWaaeaacaWGHbWaaSbaaSqaaiaad6gaaeqaaaGccaGL7bGaayzFaaaaaa@3A2E@ eta { b n } MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiWaaeaacaWGIbWaaSbaaSqaaiaad6gaaeqaaaGccaGL7bGaayzFaaaaaa@3A2F@ segidak emanik, baldintza hauek betetzen badira:

  1. { b n } MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiWaaeaacaWGIbWaaSbaaSqaaiaad6gaaeqaaaGccaGL7bGaayzFaaaaaa@3A2F@ hertsiki monotonoa bada,

  2. lim n a n + 1 a n b n + 1 b n MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9Gq=xcvk9pIe9q8qqaq=dir=f0=yqaqVeLsFr0=vr0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWfqaqaaiGacYgacaGGPbGaaiyBaaWcbaGaamOBaiabgkziUkabg6HiLcqabaGcdaWcaaqaaiaadggadaWgaaWcbaGaamOBaiabgUcaRiaaigdaaeqaaOGaeyOeI0IaamyyamaaBaaaleaacaWGUbaabeaaaOqaaiaadkgadaWgaaWcbaGaamOBaiabgUcaRiaaigdaaeqaaOGaeyOeI0IaamOyamaaBaaaleaacaWGUbaabeaaaaaaaa@4E09@ existitzen bada eta

  3. hauetako bat ere betetzen bada,

    1. lim n a n = lim n b n = 0 ; MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9Gq=xcvk9pIe9q8qqaq=dir=f0=yqaqVeLsFr0=vr0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWfqaqaaiGacYgacaGGPbGaaiyBaaWcbaGaamOBaiabgkziUkabg6HiLcqabaGccaWGHbWaaSbaaSqaaiaad6gaaeqaaOGaeyypa0ZaaCbeaeaaciGGSbGaaiyAaiaac2gaaSqaaiaad6gacqGHsgIRcqGHEisPaeqaaOGaamOyamaaBaaaleaacaWGUbaabeaakiabg2da9iaaicdacaGG7aaaaa@4FB9@

    2. lim n b n = + ; MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9Gq=xcvk9pIe9q8qqaq=dir=f0=yqaqVeLsFr0=vr0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWfqaqaaiGacYgacaGGPbGaaiyBaaWcbaGaamOBaiabgkziUkabg6HiLcqabaGccaWGIbWaaSbaaSqaaiaad6gaaeqaaOGaeyypa0Jaey4kaSIaeyOhIuQaai4oaaaa@46D9@

    3. lim n b n = ; MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9Gq=xcvk9pIe9q8qqaq=dir=f0=yqaqVeLsFr0=vr0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWfqaqaaiGacYgacaGGPbGaaiyBaaWcbaGaamOBaiabgkziUkabg6HiLcqabaGccaWGIbWaaSbaaSqaaiaad6gaaeqaaOGaeyypa0JaeyOeI0IaeyOhIuQaai4oaaaa@46E4@

Stolzen irizpideak dio berdintza hau beteko dela:

lim n a n b n = lim n a n + 1 a n b n + 1 b n MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9Gq=xcvk9pIe9q8qqaq=dir=f0=yqaqVeLsFr0=vr0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWfqaqaaiGacYgacaGGPbGaaiyBaaWcbaGaamOBaiabgkziUkabg6HiLcqabaGcdaWcaaqaaiaadggadaWgaaWcbaGaamOBaaqabaaakeaacaWGIbWaaSbaaSqaaiaad6gaaeqaaaaakiabg2da9maaxababaGaciiBaiaacMgacaGGTbaaleaacaWGUbGaeyOKH4QaeyOhIukabeaakmaalaaabaGaamyyamaaBaaaleaacaWGUbGaey4kaSIaaGymaaqabaGccqGHsislcaWGHbWaaSbaaSqaaiaad6gaaeqaaaGcbaGaamOyamaaBaaaleaacaWGUbGaey4kaSIaaGymaaqabaGccqGHsislcaWGIbWaaSbaaSqaaiaad6gaaeqaaaaaaaa@5AA2@

Adibidea Kalkula dezagun { 1 + 1 2 + + 1 n ln n } MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9Gq=xcvk9pIe9q8qqaq=dir=f0=yqaqVeLsFr0=vr0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaGadaqaamaalaaabaGaaGymaiabgUcaRmaalaaabaGaaGymaaqaaiaaikdaaaGaey4kaSIaeS47IWKaey4kaSYaaSaaaeaacaaIXaaabaGaamOBaaaaaeaaciGGSbGaaiOBaiaad6gaaaaacaGL7bGaayzFaaaaaa@46FA@ segidaren limitea.

{ b n } = { ln n } MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiWaaeaacaWGIbWaaSbaaSqaaiaad6gaaeqaaaGccaGL7bGaayzFaaGaeyypa0ZaaiWaaeaaciGGSbGaaiOBaiaad6gaaiaawUhacaGL9baaaaa@403D@ segida monotono gorakorra da eta lim n ln n = + MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9Gq=xcvk9pIe9q8qqaq=dir=f0=yqaqVeLsFr0=vr0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWfqaqaaiGacYgacaGGPbGaaiyBaaWcbaGaamOBaiabgkziUkabg6HiLcqabaGcciGGSbGaaiOBaiaad6gacqGH9aqpcqGHRaWkcqGHEisPaaa@46E1@ da. Beraz, hiru hipotesietatik bi betetzen dira, lehena eta 3.b. Kalkula dezagun 2. hipotesiko limitea:

lim n a n + 1 a n b n + 1 b n = lim n ( 1 + 1 2 + + 1 n + 1 n + 1 ) ( 1 + 1 2 + + 1 n ) ln ( n + 1 ) ln n = MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9Gq=xcvk9pIe9q8qqaq=dir=f0=yqaqVeLsFr0=vr0=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@7AB8@

lim n 1 n + 1 ln ( n + 1 n ) = lim n 1 / ( n + 1 ) 1 / n = 1. MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9Gq=xcvk9pIe9q8qqaq=dir=f0=yqaqVeLsFr0=vr0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWfqaqaaiGacYgacaGGPbGaaiyBaaWcbaGaamOBaiabgkziUkabg6HiLcqabaGcdaWcaaqaamaalaaabaGaaGymaaqaaiaad6gacqGHRaWkcaaIXaaaaaqaaiGacYgacaGGUbWaaeWaaeaadaWcaaqaaiaad6gacqGHRaWkcaaIXaaabaGaamOBaaaaaiaawIcacaGLPaaaaaGaeyypa0ZaaCbeaeaaciGGSbGaaiyAaiaac2gaaSqaaiaad6gacqGHsgIRcqGHEisPaeqaaOWaaSaaaeaadaWcgaqaaiaaigdaaeaadaqadaqaaiaad6gacqGHRaWkcaaIXaaacaGLOaGaayzkaaaaaaqaamaalyaabaGaaGymaaqaaiaad6gaaaaaaiabg2da9iaaigdacaGGUaaaaa@5CB7@

Hori ere betetzen denez, esan dezakegu segidaren limitea 1 dela.

e zenbakiaren erabilpena

e zenbakia { ( 1 + 1 n ) n } MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9Gq=xcvk9pIe9q8qqaq=dir=f0=yqaqVeLsFr0=vr0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaGadaqaamaabmaabaGaaGymaiabgUcaRmaalaaabaGaaGymaaqaaiaad6gaaaaacaGLOaGaayzkaaWaaWbaaSqabeaacaWGUbaaaaGccaGL7bGaayzFaaaaaa@418D@ segidaren bidez definitu dugu. Segida horretan aldaketa batzuk eginez, limitea ez da aldatuko edo gutxi aldatuko da. Beste aldetik, segida horretan ematen diren egoerak orokortu daitezke eta e limite bera lortu. Hauek dira emaitza nagusiak:

e = lim n ( 1 + 1 n ) n ; e = lim n ( 1 + 1 n ) n + k ; e = lim n ( 1 + 1 n + k ) n ; e k = lim n ( 1 + k n ) n ; lim n a n = 0 e = lim n ( 1 + a n ) 1 / a n ; lim n a n = ± e = lim n ( 1 + 1 a n ) a n ; MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9Gq=xcvk9pIe9q8qqaq=dir=f0=yqaqVeLsFr0=vr0=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@C14E@

Adibidea { ( n + 1 n + 3 ) n + 5 } MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9Gq=xcvk9pIe9q8qqaq=dir=f0=yqaqVeLsFr0=vr0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaGadaqaamaabmaabaWaaSaaaeaacaWGUbGaey4kaSIaaGymaaqaaiaad6gacqGHRaWkcaaIZaaaaaGaayjkaiaawMcaamaaCaaaleqabaGaamOBaiabgUcaRiaaiwdaaaaakiaawUhacaGL9baaaaa@4505@ segidaren limitea kalkulatuko dugu.

lim n ( n + 1 n + 3 ) n + 5 = lim n ( 1 2 n + 3 ) n + 3 ( 1 2 n + 3 ) 2 = e 2 × 1 = e 2 . MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9Gq=xcvk9pIe9q8qqaq=dir=f0=yqaqVeLsFr0=vr0=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@6EA0@

Cauchyren segidak

Definizioa { a n } MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiWaaeaacaWGHbWaaSbaaSqaaiaad6gaaeqaaaGccaGL7bGaayzFaaaaaa@3A2E@ segida Cauchyren segida dela esango dugu baldintza hau betetzen badu:

ε > 0 MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyiaIiIaeqyTduMaeyOpa4JaaGimaaaa@3A27@   n 0 ( ε ) / p , q n 0 ( ε ) MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaey4aIqIaamOBamaaBaaaleaacaaIWaaabeaakmaabmaabaGaeqyTdugacaGLOaGaayzkaaGaeyicI4SaeSyfHuQaai4laiabgcGiIiaadchacaGGSaGaamyCaiabgwMiZkaad6gadaWgaaWcbaGaaGimaaqabaGcdaqadaqaaiabew7aLbGaayjkaiaawMcaaaaa@49BD@    | a p a q | < ε. MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqWaaeaacaWGHbWaaSbaaSqaaiaadchaaeqaaOGaeyOeI0IaamyyamaaBaaaleaacaWGXbaabeaaaOGaay5bSlaawIa7aiabgYda8iabew7aLbaa@40CB@

Teorema Cauchyren irizpidea

MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHekaaa@375E@  multzoan { a n } MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiWaaeaacaWGHbWaaSbaaSqaaiaad6gaaeqaaaGccaGL7bGaayzFaaaaaa@3A2E@ segida konbergentea da baldin eta soilik baldin { a n } MathType@MTEF@5@5@+=feaagaart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiWaaeaacaWGHbWaaSbaaSqaaiaad6gaaeqaaaGccaGL7bGaayzFaaaaaa@3A2E@ segida Cauchyren segida bada.