Question

What is the interval of convergence of `\sum_{n=0}^{\infty} (cos x)^n`?

Answer 1

See below.

Explanation:

Using the polynomial identity

`(x^n-1)/(x-1) = 1+x+x^2+ cdots +x^(n-1)`

we have for `abs x < 1`

`lim_(n->oo) (x^n-1)/(x-1) = 1/(1-x)`

then, for `x ne k pi, k in ZZ` we have

`sum_(k=0)^oo (cos x)^k = 1/(1-cos x)`