### Maclaurin Series and Euler's Formula

Maclaurin Series: $\displaystyle f(x)=a_0+a_1x+a_2x^2+\cdots +a_nx^n+\cdots\ \ \ and \ \ a_n=\frac{f^n(0)}{n!}$

$\displaystyle e^{x}=\sum _{n=0}^{\infty }{\frac {x^{n}}{n!}}=1+x+{\frac {x^{2}}{2!}}+{\frac {x^{3}}{3!}}+\cdots +{\frac {x^{n}}{n!}}+\cdots \quad \forall x$

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

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

Euler's Formula: $\displaystyle \boxed{e^{ix}=\cos x +i\sin x}$

$\displaystyle =1+ix-\frac{x^2}{2!}-i\frac{x^3}{3!}+\frac{x^4}{4!}+i\frac{x^5}{5!}+\cdots=\Big(1-{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}-\cdots\Big)+i\Big(x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-\cdots\Big)$

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