매클로린_급수,Maclaurin_series

Difference between r1.9 and the current

@@ -1,4 +1,4 @@
[[테일러_급수,Taylor_series]]에서 중심이 0인 경우
[[테일러_급수,Taylor_series]]의 특수한 경우. 중심이 0인 경우.

$f(x)=\sum_{n=0}^{\infty}\frac{f^{(n)}(0)}{n!}x^n$
i.e.
@@ -12,6 +12,8 @@
||Maclaurin 급수 ||$\sum_{n=0}^{\infty}\frac{f^{(n)}(0)}{n!}x^n$ ||

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

@@ -22,10 +24,20 @@
$\sin x=\sum_{m=0}^{\infty}\frac{(-1)^m x^{2m+1}}{(2m+1)!}=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+\cdots$

(Kreyszig 5.1 앞부분)
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$\sin x = \sum_{n=0}^{\infty}(-1)^n\frac{x^{2n+1}}{(2n+1)!}=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\cdots\quad(-\infty<x<\infty)$
$\cos x = \sum_{n=0}^{\infty}(-1)^n\frac{x^{2n}}{(2n)!}=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\cdots\quad(-\infty<x<\infty)$
$e^x=\sum_{n=0}^{\infty}\frac{x^n}{n!}=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}+\cdots\quad(-\infty<x<\infty)$
$\ln(1+x)=\sum_{n=1}^{\infty}(-1)^{n+1}\frac{x^n}{n}=x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+\cdots\quad(-1<x\le 1)$
1이라고? chk
## tmp from https://www.youtube.com/watch?v=WYicw5Z_vKQ ; chk

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

Up: [[급수,series]]


테일러_급수,Taylor_series의 특수한 경우. 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 $\sum_{n=0}^{\infty}\frac{f^{(n)}(0)}{n!}x^n$

[edit]

잘 알려진 거듭제곱급수