Method of substitution

The method of substitution is the among the most basic technique that can be applied to infinite power series. Its use is simple to grasp and easy to master. We will first start with a couple of simple series and build up from there.

Series representation of $e^{-x}$

We know that:

$e^x=1+x+\dfrac{x^2}{2!}+\dfrac{x^3}{3!}+\dfrac{x^4}{4!}+\dfrac{x^5}{5!}+\dfrac{x^6}{6!}+O(x^8)$

$e^{-x}=1+(-x)+\dfrac{(-x)^2}{2!}+\dfrac{(-x)^3}{3!}+\dfrac{(-x)^4}{4!}+\dfrac{(-x)^5}{5!}+\dfrac{(-x)^6}{6!}+O(x^8)$

$=1-x+\dfrac{x^2}{2!}-\dfrac{x^3}{3!}+\dfrac{x^4}{4!}-\dfrac{x^5}{5!}+\dfrac{x^6}{6!}+O(x^8)$

Series representation of $e^{-x^2}$

$e^x=1+x+\dfrac{x^2}{2!}+\dfrac{x^3}{3!}+\dfrac{x^4}{4!}+\dfrac{x^5}{5!}+\dfrac{x^6}{6!}+O(x^8)$

$e^{-x^2}=1+(-x^2)+\dfrac{(-x^2)^2}{2!}+\dfrac{(-x^2)^3}{3!}+\dfrac{(-x^2)^4}{4!}+\dfrac{(-x^2)^5}{5!}+\dfrac{(-x^2)^6}{6!}+O(x^8)$

$=1-x^2+\dfrac{x^4}{2!}-\dfrac{x^6}{3!}+\dfrac{x^8}{4!}-\dfrac{x^{10}}{5!}+\dfrac{x^{12}}{6!}+O(x^{14})$

Series representation of $\dfrac{1}{\sqrt{2\pi}}e^{\frac{-x^2}{2}}$ (Normal distribution function)

$e^{-x^2}=1-x^2+\dfrac{x^4}{2!}-\dfrac{x^6}{3!}+\dfrac{x^8}{4!}-\dfrac{x^{10}}{5!}+\dfrac{x^{12}}{6!}+O(x^{14})$

$\dfrac{1}{\sqrt{2\pi}}e^{\frac{-x^2}{2}}=\dfrac{1}{\sqrt{2\pi}}\left(1-\dfrac{x^2}{2}+\dfrac{x^4}{4\cdot 2!}-\dfrac{x^6}{8\cdot 3!}+\dfrac{x^8}{16\cdot 4!}-\dfrac{x^{10}}{32\cdot 5!}+\dfrac{x^{12}}{64\cdot 6!}+O(x^{14})\right)$

Series representation of $e^{2x}$

$e^x=1+x+\dfrac{x^2}{2!}+\dfrac{x^3}{3!}+\dfrac{x^4}{4!}+\dfrac{x^5}{5!}+\dfrac{x^6}{6!}+O(x^8)$

$e^{2x}=1+2x+\dfrac{(2x)^2}{2!}+\dfrac{(2x)^3}{3!}+\dfrac{(2x)^4}{4!}+\dfrac{(2x)^5}{5!}+\dfrac{(2x)^6}{6!}+O(x^8)$

$=1+2x+\dfrac{4x^2}{2!}+\dfrac{8x^3}{3!}+\dfrac{16x^4}{4!}+\dfrac{32x^5}{5!}+\dfrac{64x^6}{6!}+O(x^8)$

$=1+2x+2x^2+\dfrac{4x^3}{3}+\dfrac{2x^4}{3}+\dfrac{4x^5}{15}+\dfrac{4x^6}{45}+O(x^8)$

Series representation of $\sin(2x)$

$\sin(x)=x-\dfrac{x^3}{3!}+\dfrac{x^5}{5!}-\dfrac{x^7}{7!}+\dfrac{x^9}{9!}-\dfrac{x^{11}}{11!}+\dfrac{x^{13}}{13!}+O(x^{15})$

$\sin(2x)=2x-\dfrac{(2x)^3}{3!}+\dfrac{(2x)^5}{5!}-\dfrac{(2x)^7}{7!}+\dfrac{(2x)^9}{9!}-\dfrac{(2x)^{11}}{11!}+\dfrac{(2x)^{13}}{13!}+O(x^{15})$

$=2x-\dfrac{8x^3}{3!}+\dfrac{32x^5}{5!}-\dfrac{128x^7}{7!}+\dfrac{512x^9}{9!}-\dfrac{2048x^{11}}{11!}+\dfrac{8192x^{13}}{13!}+O(x^{15})$

$=2x-\dfrac{4x^3}{3}+\dfrac{4x^5}{15}-\dfrac{8x^7}{315}+\dfrac{4x^9}{2835}-\dfrac{8x^{11}}{155925}+\dfrac{8x^{13}}{6081075}+O(x^{15})$

$$ \Leftrightarrow  \sum_{k=0}^{\infty}\dfrac{(-1)(2x)^{2n+1}}{(2n+1)!}$$

Series representation of $\cos(2x)$

$\cos(x)=  1-\dfrac{x^2}{2!}+\dfrac{x^4}{4!}-\dfrac{x^6}{6!}+\dfrac{x^8}{8!}-\dfrac{x^{10}}{10!}+\dfrac{x^{12}}{12!}...$

$\cos(2x)=  1-\dfrac{(2x)^2}{2!}+\dfrac{(2x)^4}{4!}-\dfrac{(2x)^6}{6!}+\dfrac{(2x)^8}{8!}-\dfrac{(2x)^{10}}{10!}+\dfrac{(2x)^{12}}{12!}...$

$=1-\dfrac{4x^2}{2!}+\dfrac{16x^4}{4!}-\dfrac{64x^6}{6!}+\dfrac{256x^8}{8!}-\dfrac{(1024x)^{10}}{10!}+\dfrac{(4096x)^{12}}{12!}+...$

$=1-2x^2+\dfrac{2x^4}{3}-\dfrac{4x^6}{45}+\dfrac{2x^8}{315}-\dfrac{4x^{10}}{14175}+\dfrac{4x^{12}}{467775}+...$

$$\Leftrightarrow  \sum_{k=0}^{\infty}\dfrac{(-1)(2x)^{2n}}{(2n)!}$$

Series representation of $\tan(2x)$

$\tan(x)=x+\dfrac{x^3}{3}+\dfrac{2x^5}{15}+\dfrac{17x^7}{315}+\dfrac{62x^9}{2835}+\dfrac{1382x^{11}}{155925}...$

$\tan(2x)=2x+\dfrac{(2x)^3}{3}+\dfrac{2(2x)^5}{15}+\dfrac{17(2x)^7}{315}+\dfrac{62(2x)^9}{2835}+\dfrac{1382(2x)^{11}}{155925}...$

$=2x+\dfrac{8x^3}{3}+\dfrac{64x^5}{15}+\dfrac{2176x^7}{315}+\dfrac{31744x^9}{2835}+\dfrac{2830336x^{11}}{155925}...$