How to evaluate Sum with Singularities?

Question

I derived equation of sum from the following problem,

$\int_{a}^{b}\sum_{n=0}^{\infty} cos^n(x)dx$.

Using the following definitions,

$cos(x)=\frac{e^{ix}+e^{-ix}}{2}$ and $(a+b)^n=\sum_{m=0}^{n}\binom {n}m a^{n-m}b^m$,

my solution to the above is,

$\sum_{n=0}^{\infty}\sum_{m=0}^{n} \binom {n}m (-i)\frac{e^{i(n-2m)}}{n-2m} $.

However, Mathematica cannot evaluate this summation because of singularities. Either my derivation is incorrect although steps are valid or I must add conditionals to this summation evaluation (if it is valid way to do). If my derivation is correct, how do I evaluate this summation,

Sum[Sum[-I*Binomial[n,m]*Exp[I*(n-2*m)/(n-2*m)),{m,0,n}],{n,0,100}]

Is there a way to avoid singularities? Like $n \neq 2m$

Answer 1

A direct approach to your problem works

sum = Sum[Cos[x]^n, {n, 0, ∞}]
(* 1/(1 - Cos[x]) *)


Integrate[sum, x]
(* -Cot[x/2] *)

Of course, the answer is only valid if the interval doesn't span a singularity (where Cos[x]==1)