Rewriting a symbolic Summation

[user3]

Consider this Summation: ∑cos^2 (∏*n / 4) limits: -N to N

when I type that on wolframAlpha I get the following:

http://www.wolframalpha.com/input/?i=summation+(1++cos(pi+n+/+2))+from+-N+to+N


I have no Idea how it was performed though.


how Can I transform this from summation form to this other algebraic form?

 

[Mark44]

Consider this Summation: ∑cos^2 (∏*n / 4) limits: -N to N

when I type that on wolframAlpha I get the following:

http://www.wolframalpha.com/input/?i=summation+(1++cos(pi+n+/+2))+from+-N+to+N


I have no Idea how it was performed though.


how Can I transform this from summation form to this other algebraic form?

Pick a value for N, and expand the summation.
Pick a different value for N, and expand the summation.
After you do this a few times, maybe you can discover a pattern.

 

[Ray Vickson]

Consider this Summation: ∑cos^2 (∏*n / 4) limits: -N to N

when I type that on wolframAlpha I get the following:

http://www.wolframalpha.com/input/?i=summation+(1++cos(pi+n+/+2))+from+-N+to+N


I have no Idea how it was performed though.


how Can I transform this from summation form to this other algebraic form?

Standard method:
$$\sum_{n=a}^{b} \cos(nw) = \frac{1}{2}\sum_{n=a}^{b} \left(e^{inw} + e^{-inw} \right)$$
This is a sum of two geometric series $\sum r^n$ with $r = e^{iw}$ and $r = e^{-iw}$.