Signal Windowing Help

Cursed

1. The problem statement, all variables and given/known data

[itex]p[n]=x[n]w[n][/itex], where [itex]w[n][/itex] is the rectangular window.



[itex]x[n]=\sum_{k=-∞}^{∞} δ[n-k][/itex]

[itex]w[n]= 1,[/itex] for [itex]-M≤n≤M;[/itex]

[itex] = 0[/itex] otherwise

1. What is [itex]X(e^{jw})[/itex]?
2. What is [itex]P(e^{jw})[/itex] when...

M=1?
M=10?

[itex]j[/itex] is the imaginary number. (it's the same as [itex]i[/itex].)

2. Relevant equations
N/A


3. The attempt at a solution

[itex]X(e^{jw})=2\pi\sum_{k=-∞}^{∞}δ[n-2\pi k][/itex]

I found [itex]X(e^{jw}[/itex] fairly easily, but I can't find [itex]P(e^{jw})[/itex] for either case of M. The answers from the book are below. Can someone tell me how I solve for this? I don't see why [itex]X(e^{jw})[/itex] is so different from [itex]P(e^{jw})[/itex]

[itex]P(e^{jw})=e^{jw}+1+e^{-jw}=1+2cos(\omega)[/itex] when M=1
[itex]P(e^{jw})=\frac{sin(21 \omega/2}{\omega/2}[/itex] when M=1