Pieaugot atvasināto polinomu skaitam, Teilora rinda tuvojas oriģinālajai funkcijai. Attēlā redzams, kā var aptuvenot sin(x) funkciju, izmantojot 1., 3.., 5., 7., 9., 11., 13. pakāpes polinomus, kad x = 0
Teilora rinda tuvojas eksponenfunkcijas grafikam
Teilora rinda matemātikā ir funkcijai, kam punktā a eksistē visu kārtu atvasinājumi, piekārtota rinda, kuras parciālsummas ir polinomi. Šo rindu 1715. gadā publicējis angļu matemātiķis Bruks Teilors (Brook Taylor).
Teilora rindu pieraksta šādi:
kur n! ir n faktoriāls un ƒ (n)(a) ir funkcijas ƒ n-tās kārtas atvasinājums punktā a.
Gadījumā, ja a = 0, tad šo rindu sauc par Maklorena rindu (nosaukta skotu matemātiķa Kolina Maklorena (Colin Maclaurin) vārdā).
Eksponentfunkcija:
![{\displaystyle e^{x}=\sum _{n=0}^{\infty }{\frac {x^{n}}{n!}}=1+x+{\frac {x^{2}}{2!}}+{\frac {x^{3}}{3!}}+\cdots \quad {\text{ visiem }}x\!}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/b12222479d08ae03d40b8b6584c4176f26f35f55)
Naturāllogaritms:
![{\displaystyle \ln(1-x)=-\sum _{n=1}^{\infty }{\frac {x^{n}}{n}}\quad {\text{, kur }}|x|<1}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/d3bf27c311f8fc7c89fd8d6645180b4e976a3cac)
![{\displaystyle \ln(1+x)=\sum _{n=1}^{\infty }(-1)^{n+1}{\frac {x^{n}}{n}}\quad {\text{, kur }}|x|<1}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/daa6645fbe2f4a45780039408270df1d2efaab99)
Ģeometriskā rinda:
![{\displaystyle {\frac {1}{1-x}}=\sum _{n=0}^{\infty }x^{n}\quad {\text{, kur }}|x|<1\!}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/ec3e5ecbca1655b6d69024fdebbccc57833f29a9)
Binomiālā rinda:
![{\displaystyle (1+x)^{\alpha }=\sum _{n=0}^{\infty }{\alpha \choose n}x^{n}\quad {\text{ visiem }}|x|<1{\text{ un kompleksajiem }}\alpha \!}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/2c4cac1e229818a1f07a0bf6d1d77d12a7187431)
ar vispārinātiem binomiālkoeficientiem
![{\displaystyle {\alpha \choose n}=\prod _{k=1}^{n}{\frac {\alpha -k+1}{k}}={\frac {\alpha (\alpha -1)\cdots (\alpha -n+1)}{n!}}}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/89ec8ca332e3268da0f2e021495164859c4d6da9)
Trigonometriskās funkcijas:
![{\displaystyle \sin x=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n+1)!}}x^{2n+1}=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-\cdots \quad {\text{ visiem }}x\!}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/29a571ecbc0084023e643cf814d25743e7208fd9)
![{\displaystyle \cos x=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n)!}}x^{2n}=1-{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}-\cdots \quad {\text{ visiem }}x\!}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/4a180bd721fd7d4d0b573759b5dced92becc78e5)
![{\displaystyle \tan x=\sum _{n=1}^{\infty }{\frac {B_{2n}(-4)^{n}(1-4^{n})}{(2n)!}}x^{2n-1}=x+{\frac {x^{3}}{3}}+{\frac {2x^{5}}{15}}+\cdots \quad {\text{, kur }}|x|<{\frac {\pi }{2}}\!}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/20751eeacfcbe6873a6cfb15f09c1606fa0cb12b)
![{\displaystyle \sec x=\sum _{n=0}^{\infty }{\frac {(-1)^{n}E_{2n}}{(2n)!}}x^{2n}\quad {\text{, kur }}|x|<{\frac {\pi }{2}}\!}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/372c087eef43aa0fed74605458d1733848297431)
![{\displaystyle \arcsin x=\sum _{n=0}^{\infty }{\frac {(2n)!}{4^{n}(n!)^{2}(2n+1)}}x^{2n+1}\quad {\text{, kur }}|x|\leq 1\!}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/5f9bd150d5301106480835322fd14c1d6d6f4061)
![{\displaystyle \arccos x={\pi \over 2}-\arcsin x={\pi \over 2}-\sum _{n=0}^{\infty }{\frac {(2n)!}{4^{n}(n!)^{2}(2n+1)}}x^{2n+1}\quad {\text{, kur }}|x|\leq 1\!}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/b63d5a47349d307fdfd9d1e3f24641c400c3a0a9)
![{\displaystyle \arctan x=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{2n+1}}x^{2n+1}\quad {\text{, kur }}|x|\leq 1,x\not =\pm i\!}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/b1834bbb5676a08bcc849e365198d414d7e6d37e)
Hiperboliskās funkcijas:
![{\displaystyle \sinh x=\sum _{n=0}^{\infty }{\frac {x^{2n+1}}{(2n+1)!}}=x+{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}+\cdots \quad {\text{ visiem }}x\!}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/482d371d32d457c7f1ea95bf6e304252a48c6591)
![{\displaystyle \cosh x=\sum _{n=0}^{\infty }{\frac {x^{2n}}{(2n)!}}=1+{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}+\cdots \quad {\text{ visiem }}x\!}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/1d4c4584cf18d4f8eaadfb96a450603875fa270b)
![{\displaystyle \tanh x=\sum _{n=1}^{\infty }{\frac {B_{2n}4^{n}(4^{n}-1)}{(2n)!}}x^{2n-1}=x-{\frac {1}{3}}x^{3}+{\frac {2}{15}}x^{5}-{\frac {17}{315}}x^{7}+\cdots \quad {\text{, kur }}|x|<{\frac {\pi }{2}}\!}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/ee97ca741d76ce6729f040edce069f14b9b3151b)
![{\displaystyle \mathrm {arcsinh} (x)=\sum _{n=0}^{\infty }{\frac {(-1)^{n}(2n)!}{4^{n}(n!)^{2}(2n+1)}}x^{2n+1}\quad {\text{, kur }}|x|\leq 1\!}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/2945751c33528fb79e530770a762751a801a5539)
![{\displaystyle \mathrm {arctanh} (x)=\sum _{n=0}^{\infty }{\frac {x^{2n+1}}{2n+1}}\quad {\text{, kur }}|x|\leq 1,x\not =\pm 1\!}](https://cdn.statically.io/img/wikimedia.org/api/rest_v1/media/math/render/svg/780394f1a15fb9234484bcfecd386a3aaef27e6d)