Interval

$of$ Convergence

$e^x=1+\frac{x}{1!}+\frac{x^2}{2!}+\frac{x^3}{3!}+$cdots$={\displaystyle \underset{n=0}{\overset{}{}}}\frac{1}{n!}{x}^n$

$cosx=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+$cdots$={\displaystyle \underset{n=0}{\overset{}{}}}\frac{(-1{)}^n}{(2n)!}{x}^{2n}$

$sinx=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+$cdots$={\displaystyle \underset{n=0}{\overset{}{}}}\frac{(-1{)}^n}{(2n+1)!}{x}^{2n+1}$

$ta{n}^{-1}x=x-\frac{x^3}{3}+\frac{x^5}{5}-\frac{x^7}{7}+$cdots$={\displaystyle \underset{n=0}{\overset{}{}}}\frac{(-1{)}^n}{2n+1}{x}^{2n+1}$

$coshx=1+\frac{x^2}{2!}+\frac{x^4}{4!}+\frac{x^6}{6!}+$cdots$={\displaystyle \underset{n=0}{\overset{}{}}}\frac{1}{(2n)!}{x}^{2n}$

$sinhx=x+\frac{x^3}{3!}+\frac{x^5}{5!}+\frac{x^7}{7!}+$cdots$={\displaystyle \underset{n=0}{\overset{}{}}}\frac{1}{(2n+1)!}{x}^{2n+1}$

$ln(1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+$cdots$={\displaystyle \underset{n=1}{\overset{}{}}}\frac{(-1{)}^{n+1}}{n}{x}^n$

$\frac{1}{1-x}=1+x+x^2+x^3+$cdots$={\displaystyle \underset{n=0}{\overset{}{}}}{x}^n-,-,-,-1,1-,-,(-1,1]$

$(-1,1)$