If $ \displaystyle \sum_{n \equal{} 0}^{\infty} \cos^{2n} \theta \equal{} 5$, what is the value of $ \cos{2\theta}$?

$ \textbf{(A)}\  \frac15 \qquad \textbf{(B)}\  \frac25 \qquad \textbf{(C)}\  \frac {\sqrt5}{5}\qquad \textbf{(D)}\  \frac35 \qquad \textbf{(E)}\ \frac45$
1. We start with the given infinite series:
   \[
   \sum_{n=0}^{\infty} \cos^{2n} \theta = 5
   \]
   This is a geometric series with the first term \(a = 1\) and common ratio \(r = \cos^2 \theta\).

2. The sum of an infinite geometric series is given by:
   \[
   S = \frac{a}{1 - r}
   \]
   Substituting \(a = 1\) and \(r = \cos^2 \theta\), we get:
   \[
   \frac{1}{1 - \cos^2 \theta} = 5
   \]

3. Solving for \(\cos^2 \theta\):
   \[
   1 - \cos^2 \theta = \frac{1}{5}
   \]
   \[
   \cos^2 \theta = 1 - \frac{1}{5} = \frac{4}{5}
   \]

4. We need to find \(\cos 2\theta\). Using the double-angle identity for cosine:
   \[
   \cos 2\theta = 2\cos^2 \theta - 1
   \]
   Substituting \(\cos^2 \theta = \frac{4}{5}\):
   \[
   \cos 2\theta = 2 \left( \frac{4}{5} \right) - 1 = \frac{8}{5} - 1 = \frac{8}{5} - \frac{5}{5} = \frac{3}{5}
   \]

The final answer is \(\boxed{\frac{3}{5}}\)