# Max of sum of sines

1. Apr 23, 2014

### ekkilop

Hi!

Consider the function

$\frac{d^n}{dx^n} \sum_{k=1}^m \sin{kx}, \quad 0 \leq x \leq \pi/2$.

If $n$ is odd this function attains its largest value, $\sum_{k=1}^m k^n$ at $x=0$. But what about if $n$ is even? Where does the maximum occur and what value does it take?

Any help is much appreciated. Thank you!

Last edited: Apr 23, 2014
2. Apr 23, 2014

### Simon Bridge

Please make some attempt at the problem when you ask for help.
i.e. how would you normally go about finding the maximum?
Can you prove the statement about when m is odd?