taylor_series
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Taylor series
Taylor's series of $e^x$
일반화된 설명은 너무 복잡하고 아래와 같이 생각해보자
$e^x = \displaystyle \sum_{k=0}^\infty {\frac{x^k}{k!}}$ 이라고 하면
\begin{eqnarray*} e^x & = & \displaystyle \sum_{k=0}^\infty {\frac{x^k}{k!}} \\ & = & 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + . . . \\ \end{eqnarray*}
실제 계산을 해보면
Terms | |
---|---|
$1 + 2$ | $3$ |
$1 + 2 + \displaystyle \frac{2^2}{2!}$ | $5$ |
$1 + 2 + \displaystyle \frac{2^2}{2!} + \displaystyle \frac{2^3}{3!} $ | $6.333...$ |
$1 + 2 + \displaystyle \frac{2^2}{2!} + \displaystyle \frac{2^3}{3!} + \displaystyle \frac{2^4}{4!} $ | $7$ |
$1 + 2 + \displaystyle \frac{2^2}{2!} + \displaystyle \frac{2^3}{3!} + \displaystyle \frac{2^4}{4!} + \displaystyle \frac{2^5}{5!} $ | $7.2666...$ |
$1 + 2 + \displaystyle \frac{2^2}{2!} + \displaystyle \frac{2^3}{3!} + \displaystyle \frac{2^4}{4!} + \displaystyle \frac{2^5}{5!} + \displaystyle \frac{2^6}{6!} $ | $7.3555...$ |
$1 + 2 + \displaystyle \frac{2^2}{2!} + \displaystyle \frac{2^3}{3!} + \displaystyle \frac{2^4}{4!} + \displaystyle \frac{2^5}{5!} + \displaystyle \frac{2^6}{6!} + \displaystyle \frac{2^7}{7!} $ | $7.3809...$ |
$1 + 2 + \displaystyle \frac{2^2}{2!} + \displaystyle \frac{2^3}{3!} + \displaystyle \frac{2^4}{4!} + \displaystyle \frac{2^5}{5!} + \displaystyle \frac{2^6}{6!} + \displaystyle \frac{2^7}{7!} + \displaystyle \frac{2^8}{8!} $ | $7.3873...$ |
taylor_series.1605885855.txt.gz · Last modified: 2020/11/21 00:24 by hkimscil