Maclaurin series for some common functions .

A Maclaurin series is a Taylor series expansion of a function about 0
\( \displaystyle f(x)=f(0)+f^{1}(0)x+\displaystyle \frac{f^{2}(0)}{2!}x^2+\displaystyle \frac{f^{3}(0)}{3!}x^3+...+\displaystyle \frac{f^{n}(0)}{n!}x^{n}+....\)

Maclaurin series for some common functions are given below

\(\bullet \;\;\displaystyle \frac{1}{(1-x)}=1+x+x^2+x^3+x^4+x^5+ \cdots+x^n \;\;for\;\;-1 <x<1 \)

\(\bullet\;\;\displaystyle e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}+...+\frac{x^n}{n!} \;\; -\infty < x < \infty\)

\(\bullet \;\;\displaystyle \ln(1+x)=\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n}x^n \;\;for\;\; -1 < x \leq 1\)

\(\bullet \;\;\displaystyle \ln\left ( \frac{1+x}{1-x} \right )=\sum_{n=1}^{\infty}\frac{2}{2n-1}x^{2n-1} \;\;for\;\; -1 < x < 1\)

\(\bullet \;\;\displaystyle \sin x=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-\cdots \sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n+1)!}}x^{2n+1} \; \;\; -\infty < x < \infty\)

\(\bullet\;\;\displaystyle \cos x=1-{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}-\cdots \sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n)!}}x^{2n}\;\;\;\; -\infty < x < \infty\)

\(\bullet \;\;\displaystyle \tan x=\sum _{n=1}^{\infty }{\frac {B_{2n}(-4)^{n}\left(1-4^{n}\right)}{(2n)!}}x^{2n-1}=x+{\frac {x^{3}}{3}}+{\frac {2x^{5}}{15}}+\cdots \;\;\forall |x| < \frac {\pi }{2}\)

\(\bullet \;\;\displaystyle \sec x=\sum _{n=0}^{\infty }{\frac {(-1)^{n}E_{2n}}{(2n)!}}x^{2n}=1+{\frac {x^{2}}{2}}+{\frac {5x^{4}}{24}}+\cdots \;\;\forall |x| < \frac {\pi }{2} \)

\(\bullet \;\;\displaystyle \sin^{-1} x=x+{\frac {x^{3}}{6}}+{\frac {3x^{5}}{40}}+\cdots \sum _{n=0}^{\infty }{\frac {(2n)!}{4^{n}(n!)^{2}(2n+1)}}x^{2n+1} \;\; \forall |x|\leq 1\)

\(\bullet\;\; \displaystyle \ \cos^{-1} x={\frac {\pi }{2}}-\sum _{n=0}^{\infty }{\frac{(2n)!}{4^{n}(n!)^{2}(2n+1)}}x^{2n+1}\;\; \forall |x|\leq 1\)

\(\bullet\;\; \displaystyle \tan^{-1}x=\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{2n-1}x^{2n-1}\)

\(\bullet\;\; \displaystyle \cot^{-1}x=\frac{\pi}{2}-\sum_{n=0}^{\infty}\frac{(-1)^{n}}{(2n+1)}x^{2n+1}\)

\(\bullet \;\;\displaystyle \sinh x= x+\frac{1}{6}x^3+\frac{1}{120}x^5+\frac{1}{5040}x^7+\frac{1}{362880}x^9+\cdots \sum_{n=0}^{\infty}\frac{1}{(2n+1)!}x^{2n+1}\)

\(\bullet\;\; \displaystyle \cosh x=1+\frac{1}{2x^2}+\frac{1}{24}x^4+\frac{1}{720}x^6+\frac{1}{40,320}x^8+\cdots \sum_{n=0}^{\infty}\frac{1}{(2n)!}x^{2n}\)

\(\bullet \;\;\displaystyle \tanh x=x-\frac{1}{3}x^3+\frac{2}{15}x^5-\frac{17}{315}x^7+\frac{62}{2835}x^9+\cdots\)

or

\(\displaystyle \sum_{n=1}^{\infty}(\frac{2^{2n}(2^{2n}-1)B_{2n}}{2n!}x^{2n-1} \)