Power Expansion (Complex variables)

[tmlfan_17]

Homework Statement

Use the power series for e^z and the def. of sin(z) to check that
sum ((-1)^k z^(2 k+1))/((2 k+1)!)

Homework Equations

The Attempt at a Solution

I apologize, but I am not particularly good with latex. Therefore, I attached a picture of my solution thus far. I've tried many methods, but this is where I get stuck and I can't seem to get sin(z) to equal its power expansion. Any help would be very much appreciated.

 

[Saitama]

So you have:
$$\frac{1}{2i}\sum_{n=0}^{\infty} \frac{z^n}{n!}\left(i^n-(-i)^n\right)$$
Clearly, if $n$ is even, $i^n-(-i)^n=0$. Can you figure out what happens if $n$ is odd i.e $n$ is of the form $2k+1$?

 

[tmlfan_17]

Yes. Thank you sir!

 

[Saitama]

Yes. Thank you sir!

Glad to help but please don't call me sir, I am a student myself.