Summation of the given series Is there anyway to find the sum of:
$\cos(A)+\cos^2(2A)+\cos^3(3A)+....$ upto 'n' terms.

Actually original question was to find sum of :
$\cos(A)+\cos(2A)+\cos(3A)+...$ upto 'n' terms and I found it with help of De-Movier's Theorem which also gave me sum of $\sin(A)+\sin(2A)+.....$. But now I am thinking can we also find sum of $\cos(A)+\cos^2(2A)+\cos^3(3A)+....$ upto 'n' terms by some method?
Use the fact $$\cos^k ( nA) =\left(\frac{e^{inA} +e^{-inA}}{2}\right)^k$$
Use the fact $$\cos^k ( nA) =\left(\frac{e^{inA} +e^{-inA}}{2}\right)^k$$