Question

What is the infinite sum of sin^2[wn]?

Answer 1

This is actually for my signals and systems class, but I am having a lot of difficulty remembering all the stuff about sequences and series from back when I took calc 2.

Here is how I set it up initially:\[\sum_{n=-\infty}^{\infty}\sin^2[w_0n] = \sum_{n=0}^{\infty}\sin^2[w_0n] - \sum_{n= 0}^{-\infty}\sin^2[w_0n] \]I'm not sure if that is the proper way to break up a sum like this, but my professor said that pretty much every one of these problems should work out to either an arithmetic or geometric series, so I should need n=0 initially right?

Answer 2

I'm not sure if I can evaluate it using the normal arithmetic series solution of \[\sum_{n = 0}^{N} \sin^2[w_0n] = \frac{N(\sin^2[w_0]+ \sin^2[w_0N])}{2}\]or if I should convert sin using euler's identity and work with complex exponentials..