I would have done a direct comparison with to show convergence.
find the values for which the series converges. FInd the sum of the series for those values of x. sum {sn} from n = 0 to infinity. of (cos x)^n/(2^n).
my work: sum {sn} from k =1 to infinity where n = k-1. of [(cos x)/2]^ (k-1)
so the sum is 1/(1-(cosx)/2) converges for all x.
is this right?