Definition. The Taylor series of $ƒ\left(x\right)$, that is infinitely differentiable at a real or complex number $a$, is the power series

$ƒ\left(x\right)=ƒ\left(a\right)+{ƒ\left(a\right)}^{\prime }(x-a)+\frac{{{{ƒ}^{}\left(a\right)}^{\prime }}^{\prime }}{2!}{(x-a)}^2+\cdots +\frac{{ƒ}^n\left(a\right)}{n!}{(x-a)}^n$

Maclaurin Series: Taylor series at $a=0$:

$ƒ\left(x\right)=ƒ\left(0\right)+{ƒ}^{\prime }\left(0\right)x+\frac{{ƒ}^{\prime \prime }\left(0\right)}{\mathrm{2!}}{x}^2+\frac{{ƒ}^{\prime ″}\left(0\right)}{\mathrm{3!}}{x}^3+\cdots ={\sum }_{n=0}^{\infty }(\frac{{ƒ}^n\left(0\right)}{n!}{x}^n)$

Taylor Series of known functions.

$\frac{1}{1-x}=1+x+x^2+x^3+\cdots ={\sum }_{n=0}^{\infty }{x}^n$

${ℇ}^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\cdots ={\sum }_{n=0}^{\infty }\frac{x^n}{n!}$

$\cos x=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\cdots ={\sum }_{n=0}^{\infty }({(-1)}^n\cdot \frac{x^{2n}}{(2n)!})$

$\sin x=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+\cdots ={\sum }_{n=0}^{\infty }({(-1)}^{n+1}\cdot \frac{x^{2n+1}}{(2n+1)!})$

$\ln (1+x)=x-\frac{x^2}{2!}+\frac{x^3}{3!}-\frac{x^4}{4!}+\frac{x^5}{5!}-\frac{x^6}{6!}+\frac{x^7}{7!}+\cdots ={\sum }_{n=0}^{\infty }({(-1)}^{n+1}\cdot \frac{x^n}{n!})$

$\frac{1}{\tan x}=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+\cdots ={\sum }_{n=0}^{\infty }({(-1)}^n\frac{x^{2n+1}}{(2n+1)!})$

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