calculus - Find the sum of infinite series when ever the series converges.

Hi I am trying to find the sum of this infinite series whenever it converges. I have tried the common ratio technique but my work doesn't match the answers. I would appreciate any help with explanation.

$$\sum_{k=1}^\infty (2\cos^2 \theta)^k $$

Answer

this is a geometric series. So answer is
$$
\frac{2\cos^2 \theta}{1-2\cos^2 \theta}
$$
whenever
$$
\cos^2 \theta <\frac12
$$