بسط تیلور

${\displaystyle f(x)=f(x_0)+\frac{f^{\prime }(x_0)(x-x_0)}{1!}+\frac{f^{″}(x_0)(x-x_0{)}^2}{2!}+\frac{f^{‴}(x_0)(x-x_0{)}^3}{3!}+...}$

${\displaystyle f(x)=\sum _{n=0}^{\mathrm{\infty }}\frac{f^{(n)}(x_0)}{n!}(x-x_0{)}^n}$

${\displaystyle f(x)\sim P(x)=f(a)+\frac{f^{\prime }(a)}{1!}(x-a)+\frac{f^{″}(a)}{2!}(x-a{)}^2+\frac{f^{(3)}(a)}{3!}(x-a{)}^3+\cdots .}$

${\displaystyle f(x)\sim P(x)=\sum _{n=0}^{\mathrm{\infty }}\frac{f^{(n)}(a)}{n!}(x-a{)}^n}$

${\displaystyle \sin (x)=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\cdots \text{ for all }x}$

${\displaystyle \sin (x)\sim x-\frac{x^3}{3!}}$

${\displaystyle f(x)=e^{2x}}$

${\displaystyle e^{2x}=\frac{(1)}{(e^2)}+\frac{(2)(x+1)}{(1!)(e^2)}+\frac{(2^2)(x+1{)}^2}{(2!)(e^2)}+\frac{(2^3)(x+1{)}^3}{(3!)(e^2)}+...}$

${\displaystyle e^{2x}=\sum _{n=0}^{\mathrm{\infty }}\frac{(2{)}^{(n)}}{((n)!e^2)}(x+1{)}^n}$

${\displaystyle e^x=\sum _{n=0}^{\mathrm{\infty }}\frac{x^n}{n!}=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\cdots \text{ for all }x}$

${\displaystyle \frac{1-x^{m+1}}{1-x}=\sum _{n=0}^m{x}^n\text{ for }x\ne 1\text{ and }m\in {\mathbb{N}}_0}$

${\displaystyle \frac{1}{1-x}=\sum _{n=0}^{\mathrm{\infty }}{x}^{-n}\text{ for }|x|>1}$

${\displaystyle \frac{1}{(1-x{)}^2}=\sum _{n=1}^{\mathrm{\infty }}n{x}^{n-1}\text{ for }|x|>1}$

${\displaystyle \sqrt{1+x}=\sum _{n=0}^{\mathrm{\infty }}\frac{(-1{)}^n(2n)!}{(1-2n)(n!{)}^2(4^n)}{x}^n=1+\frac{1}{2}x-\frac{1}{8}{x}^2+\frac{1}{16}{x}^3-\frac{5}{128}{x}^4+\dots \text{ for }|x|\le 1}$

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

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

${\displaystyle \arcsin x=\sum _{n=0}^{\mathrm{\infty }}\frac{(2n)!}{4^n(n!{)}^2(2n+1)}{x}^{2n+1}\text{ for }|x|\le 1}$

${\displaystyle \arccos x=\frac{\pi }{2}-\arcsin x=\frac{\pi }{2}-\sum _{n=0}^{\mathrm{\infty }}\frac{(2n)!}{4^n(n!{)}^2(2n+1)}{x}^{2n+1}\text{ for }|x|\le 1}$

${\displaystyle \arctan x=\sum _{n=0}^{\mathrm{\infty }}\frac{(-1{)}^n}{2n+1}{x}^{2n+1}\text{ for }|x|\le 1}$

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

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

${\displaystyle \mathrm{a}\mathrm{r}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}(x)=\sum _{n=0}^{\mathrm{\infty }}\frac{(-1{)}^n(2n)!}{4^n(n!{)}^2(2n+1)}{x}^{2n+1}\text{ for }|x|\le 1}$