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Dragon Notes

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  UNDER CONSTRUCTION

Power Series


\(\ds \begin{align} & \bb{1} & e^x &=&& 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + ... &&= \sum_{n=0}^{\infty}\frac{x^n}{n!} && x\in (-1,1)\\ & \bb{2} & \frac{1}{1-x} &=&& 1 + x + x^2 + x^3 + x^4 + ... &&= \sum_{n=0}^{\infty}x^n && \\ & \bb{3} & \ln{1+x} &=&& x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+\frac{x^5}{5}-... &&= \sum_{n=1}^{\infty}(-1)^{n+1}\frac{x^n}{n} && x\in (-1,1] \\ & \bb{4} & \Sin{x} &=&& x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \frac{x^9}{9!} - ... &&= \sum_{n=0}^{\infty}(-1)^n \frac{x^{2n+1}}{(2n+1)!} & &\\ & \bb{5} & \Cos{x} &=&& 1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\frac{x^8}{8!}-... &&= \sum_{n=0}^{\infty}\frac{x^{2n}}{(2n)!} && \\ & \bb{6} & \sinh{(x)} &=&& x + \frac{x^3}{3!} + \frac{x^5}{5!} + \frac{x^7}{7!} + \frac{x^9}{9!} + ... &&= \sum_{n=0}^{\infty} \frac{x^{2n+1}}{(2n+1)!} && \\ & \bb{7} & \cosh{(x)} &=&& 1 + \frac{x^2}{2!} + \frac{x^4}{4!} + \frac{x^6}{6!} + \frac{x^8}{8!} + ... &&= \sum_{n=0}^{\infty} \frac{x^{2k}}{(2k)!} \\ & \bb{8} & \atan{x} &=&& x-\frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \frac{x^9}{9} - ... &&= \sum_{n=0}^{\infty}(-1)^n \frac{x^{2n+1}}{2n+1} && x\in [-1,1] \end{align}\)

Longer series

\(\ds \begin{align} && \bb{1'}\ (a+x)^t &= a^t + txa^{t-1} + \frac{1}{2}(t-1)tx^2a^{t-2}+\frac{1}{6}(t-2)(t-1)tx^3a^{t-3} &&= \sum_{n=0}^{\infty}\frac{t^n\t{log}^n(a+x)}{n!} && \\ && & &&= \zisum{n}{t \choose n}(-1+a+x)^n && |-1+a+x|<1 \\ && & &&= \zisum{n}{t \choose n}(-1+a+x)^{t-n} && |-1+a+x|>1 \end{align}\)


Dragon Notes,   Est. 2018     About

By OverLordGoldDragon