The Taylor Series for \( \sin x \) centered at \( x=\frac{\pi}3 \)
| \( g(x)=\sin x \) | \( g^{\prime}\left(\frac{\pi}3 \right)=\frac{1}2 \) |
| \( g^{\prime}(x)=\cos x \) | \( g^{\prime}\left(\frac{\pi}3 \right)=\frac{1}2 \) |
| \( g^{\prime\prime}(x)=-\sin x \) | \( g^{\prime\prime}\left(\frac{\pi}3 \right)=\frac{-\sqrt 3}2 \) |
| \( g^{\prime\prime\prime}(x)=-\cos x \) | \( g^{\prime\prime\prime}\left(\frac{\pi}3 \right)=\frac{-1}2 \) |
| \( g^{(4)} (x)=\sin x \) | \( g^{(4)}\left(\frac{\pi}3 \right)=\frac{\sqrt 3}2 \) |
| \(\sin x =\frac {\sqrt 3}{2 \cdot 0!}\left(x-\frac{\pi}3 \right)^{0}+\frac {1}{2\cdot 1!}\left(x-\frac {\pi}3 \right)^{1}-\frac {\sqrt 3}{2 \cdot 2!}\left(x-\frac {\pi}3 \right)^{2}-\frac {1}{2 \cdot 3!} \left(x- \frac{\pi}3 \right)^{3}+\frac {\sqrt 3}{2 \cdot 4!} \left(x- \frac{\pi}3 \right)^{4}+ \cdots\) |
| \(\sin x =\frac{\sqrt 3}{2} \left(1-\frac{1}{2!} \left(x-\frac{\pi}3 \right)^{2}+\frac{1}{4!} \left(x-\frac{\pi}3 \right)^{4}- \cdots \right)+ \frac{1}2 \left( \left( x – \frac{\pi}3 \right) – \frac{1}{3!} \left( x – \frac{\pi}3 \right)^{3} + \frac{1}{5!} \left( x – \frac{\pi}3 \right)^{5} + \cdots \right)\) |
| \(\sin x=\frac {\sqrt 3}2 \sum_{n=0}^{\infty} \frac{ \left(-1 \right)^n}{ \left(2n \right)!} \left( x – \frac {\pi}3 \right)^{2n}+ \frac {1}2 \sum_{n=0}^{\infty} \frac { \left(-1 \right)^{n}}{ \left(2n+1 \right)!} \left( x – \frac{\pi}3 \right)^{2n+1}\) |