[[테일러_급수,Taylor_series]]에서 중심이 0인 경우

$f(x)=\sum_{n=0}^{\infty}\frac{f^{(n)}(0)}{n!}x^n$
i.e.
$f(x)=\frac{f(0)}{0!}x^0+\frac{f^{\prime}(0)}{1!}x^1+\frac{f^{\prime\prime}(0)}{2!}x^2+\;\cdots$
i.e.
$f(x)=f(0)+f^{\prime}(0)x+\frac{f^{\prime\prime}(0)}{2!}x^2+\frac{f^{(3)}(0)}{3!}x^3+\;\cdots\;+\frac{f^{(n)}(0)}{n!}x^n+\;\cdots$


----
||Taylor 급수 ||$\sum_{n=0}^{\infty}{\frac{f^{(n)}(x_{0})}{n!}(x-x_{0})^{n}}$ ||
||Maclaurin 급수 ||$\sum_{n=0}^{\infty}\frac{f^{(n)}(0)}{n!}x^n$ ||

= 잘 알려진 거듭제곱급수 =
$\frac1{1-x}=\sum_{m=0}^{\infty}x^m=1+x+x^2+x^3+\cdots$
 $|x|<1,$ [[기하급수,geometric_series|등비급수]].

$e^x=\sum_{m=0}^{\infty}\frac{x^m}{m!}=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\cdots$

$\cos x=\sum_{m=0}^{\infty}\frac{(-1)^m x^{2m}}{(2m)!}=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\cdots$

$\sin x=\sum_{m=0}^{\infty}\frac{(-1)^m x^{2m+1}}{(2m+1)!}=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+\cdots$

(Kreyszig 5.1 앞부분)

----
http://mathworld.wolfram.com/MaclaurinSeries.html

Up: [[급수,series]]