| \begin{eqnarray*} f(x)&=&\sin x\qquad x_0=0\\ f(x)&\approx&\sin 0+x\cos0=x\\ f'(x)&=&\cos x\rightarrow f''(x)=-\sin x\rightarrow f'''(x)=-\cos x\\ & & \qquad \; \rightarrow f''''(x)=\sin x=f(x)\\ \rightarrow f^{n}(0)&=&\left\{\begin{array}{lcl}1&\mbox{if}&n=2k+1,\;k\mbox{ even}\\-1&\mbox{if}&n=2k+1,\;k\mbox{ odd}\\0&\mbox{if}&n=2k\end{array}\right.\\ \rightarrow f(x)&=&\sum_{n=0}^\infty\frac{(-1)^n}{(2n+1)!}x^{2n+1}=x-\frac{1}{3!}x^3+\frac{1}{5!}x^5-\ldots\end{eqnarray*} |
With frame
Taylor series and their application