Sum of series $2^{2m}$ How to sum $2^{2m}$ where $m$ varies from $0$ to $n$?

If you really mean $n$ varies from $0$ to $n$ then the answer is
$\sum _{n=0}^n 2^{2 m}=2^{2 m} (1 + n)$
Otherwise if $m$ goes from $0$ to $n$ its just a geometric series ( http://mathworld.wolfram.com/GeometricSeries.html )
$\sum _{m=0}^n 2^{2 m}=\frac{1}{3} \left(2^{2 n+2}-1\right)$
