# Power Expansion (Complex variables)

## 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)!)

## 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.

#### Attachments

• Scan 1.jpg
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## Answers and Replies

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##?

Yes. Thank you sir!

Yes. Thank you sir!

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