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:

${\displaystyle \sum _{n=0}^{\infty }{\frac {f^{(n)}(a)}{n!))\,(x-a)^{n}=f(a)+{\frac {f'(a)}{1!))(x-a)+{\frac {f''(a)}{2!))(x-a)^{2}+{\frac {f^{(3)}(a)}{3!))(x-a)^{3}+\cdots ,}$

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).

Dažu funkciju izvirzījumi Maklorena rindā

Eksponentfunkcija:

${\displaystyle e^{x}=\sum _{n=0}^{\infty }{\frac {x^{n)){n!))=1+x+{\frac {x^{2)){2!))+{\frac {x^{3)){3!))+\cdots \quad {\text{ visiem ))x\!}$

Naturāllogaritms:

${\displaystyle \ln(1-x)=-\sum _{n=1}^{\infty }{\frac {x^{n)){n))\quad {\text{, kur ))|x|<1}$

${\displaystyle \ln(1+x)=\sum _{n=1}^{\infty }(-1)^{n+1}{\frac {x^{n)){n))\quad {\text{, kur ))|x|<1}$

Ģeometriskā rinda:

${\displaystyle {\frac {1}{1-x))=\sum _{n=0}^{\infty }x^{n}\quad {\text{, kur ))|x|<1\!}$

Binomiālā rinda:

${\displaystyle (1+x)^{\alpha }=\sum _{n=0}^{\infty }{\alpha \choose n}x^{n}\quad {\text{ visiem ))|x|<1{\text{ un kompleksajiem ))\alpha \!}$

ar vispārinātiem binomiālkoeficientiem

${\displaystyle {\alpha \choose n}=\prod _{k=1}^{n}{\frac {\alpha -k+1}{k))={\frac {\alpha (\alpha -1)\cdots (\alpha -n+1)}{n!))}$

Trigonometriskās funkcijas:

${\displaystyle \sin x=\sum _{n=0}^{\infty }{\frac {(-1)^{n)){(2n+1)!))x^{2n+1}=x-{\frac {x^{3)){3!))+{\frac {x^{5)){5!))-\cdots \quad {\text{ visiem ))x\!}$

${\displaystyle \cos x=\sum _{n=0}^{\infty }{\frac {(-1)^{n)){(2n)!))x^{2n}=1-{\frac {x^{2)){2!))+{\frac {x^{4)){4!))-\cdots \quad {\text{ visiem ))x\!}$

${\displaystyle \tan x=\sum _{n=1}^{\infty }{\frac {B_{2n}(-4)^{n}(1-4^{n})}{(2n)!))x^{2n-1}=x+{\frac {x^{3)){3))+{\frac {2x^{5)){15))+\cdots \quad {\text{, kur ))|x|<{\frac {\pi }{2))\!}$

${\displaystyle \sec x=\sum _{n=0}^{\infty }{\frac {(-1)^{n}E_{2n)){(2n)!))x^{2n}\quad {\text{, kur ))|x|<{\frac {\pi }{2))\!}$

${\displaystyle \arcsin x=\sum _{n=0}^{\infty }{\frac {(2n)!}{4^{n}(n!)^{2}(2n+1)))x^{2n+1}\quad {\text{, kur ))|x|\leq 1\!}$

${\displaystyle \arccos x={\pi \over 2}-\arcsin x={\pi \over 2}-\sum _{n=0}^{\infty }{\frac {(2n)!}{4^{n}(n!)^{2}(2n+1)))x^{2n+1}\quad {\text{, kur ))|x|\leq 1\!}$

${\displaystyle \arctan x=\sum _{n=0}^{\infty }{\frac {(-1)^{n)){2n+1))x^{2n+1}\quad {\text{, kur ))|x|\leq 1,x\not =\pm i\!}$

Hiperboliskās funkcijas:

${\displaystyle \sinh x=\sum _{n=0}^{\infty }{\frac {x^{2n+1)){(2n+1)!))=x+{\frac {x^{3)){3!))+{\frac {x^{5)){5!))+\cdots \quad {\text{ visiem ))x\!}$

${\displaystyle \cosh x=\sum _{n=0}^{\infty }{\frac {x^{2n)){(2n)!))=1+{\frac {x^{2)){2!))+{\frac {x^{4)){4!))+\cdots \quad {\text{ visiem ))x\!}$

${\displaystyle \tanh x=\sum _{n=1}^{\infty }{\frac {B_{2n}4^{n}(4^{n}-1)}{(2n)!))x^{2n-1}=x-{\frac {1}{3))x^{3}+{\frac {2}{15))x^{5}-{\frac {17}{315))x^{7}+\cdots \quad {\text{, kur ))|x|<{\frac {\pi }{2))\!}$

${\displaystyle \mathrm {arcsinh} (x)=\sum _{n=0}^{\infty }{\frac {(-1)^{n}(2n)!}{4^{n}(n!)^{2}(2n+1)))x^{2n+1}\quad {\text{, kur ))|x|\leq 1\!}$

${\displaystyle \mathrm {arctanh} (x)=\sum _{n=0}^{\infty }{\frac {x^{2n+1)){2n+1))\quad {\text{, kur ))|x|\leq 1,x\not =\pm 1\!}$