$$f(x)=sin x = \sum_{n=0}^{\infty} \frac{f^n (\frac{\pi}{2})}{2n!} (x-\frac{\pi}{2})^{2n}$$
It's an alternating series, how do I show that in my sum equation

Question

anonymous · Answer

$$(-1)^n$$?

anonymous · Answer

a= pi/2

anonymous · Answer

$$
\sum _{n=0}^\infty \frac{(-1)^n \left(x-\frac{\pi
   }{2}\right)^{2 n}}{(2 n)!}
$$

anonymous · Answer

yay!...so (-1)^n for alternating series

anonymous · Answer

wait lets compare
$$\sum _{n=0}^\infty \frac{(-1)^n \left(x-\frac{\pi
   }{2}\right)^{2 n}}{(2 n)!}$$
$$\sum_{n=0}^{\infty} \frac{f^n (\frac{\pi}{2})}{2n!} (x-\frac{\pi}{2})^{2n}$$
so what happened to f^n (pi/2)...is that not necessary to write?

anonymous · Answer

Here are the first 21 derivatives of sin(x)
$$

\begin{array}{cc}
 0 & \sin (x) \
 1 & \cos (x) \
 2 & -\sin (x) \
 3 & -\cos (x) \
 4 & \sin (x) \
 5 & \cos (x) \
 6 & -\sin (x) \
 7 & -\cos (x) \
 8 & \sin (x) \
 9 & \cos (x) \
 10 & -\sin (x) \
 11 & -\cos (x) \
 12 & \sin (x) \
 13 & \cos (x) \
 14 & -\sin (x) \
 15 & -\cos (x) \
 16 & \sin (x) \
 17 & \cos (x) \
 18 & -\sin (x) \
 19 & -\cos (x) \
 20 & \sin (x) \
\end{array}

$$

anonymous · Answer

Those derivatives evaluated at $\pi/2 $ 
$$
\begin{array}{cc}
 0 & 1 \
 1 & 0 \
 2 & -1 \
 3 & 0 \
 4 & 1 \
 5 & 0 \
 6 & -1 \
 7 & 0 \
 8 & 1 \
 9 & 0 \
 10 & -1 \
 11 & 0 \
 12 & 1 \
 13 & 0 \
 14 & -1 \
 15 & 0 \
 16 & 1 \
 17 & 0 \
 18 & -1 \
 19 & 0 \
 20 & 1 \
\end{array}

$$

anonymous · Answer

yep I made a table
n        f^n x      f^n (pi/2)
0        sinx          1
1        cosx         0
2       -sinx          -1
3        -cosx        0
4         sinx          1

anonymous · Answer

but in my final summation notation   do I not need to say f^(n) (pi/2)

anonymous · Answer

No

anonymous · Answer

ok so we did this problem together yesterday and the final answer then is 
$$f(x)=\frac{1}{x}=\sum_{n=0}^{\infty} \frac{(x+3)^n}{n!}$$

anonymous · Answer

for a=-3

anonymous · Answer

and what?

anonymous · Answer

oh I see you were trying to explain the series to me that's why you left the f^n (-3) in the summation notation. Makes sense now