I'm trying to find a formula for the series below without success. I've tried to use: $\sum_{k=1}^n ar^{k-1} = \frac{a(1-r^n)}{1-r}$ but my $n$ messes things up. I have trouble understanding what is the first term and the ratio in my series. Any help is appreciated.
$\sum_{i=0}^k ((\frac{1}{2})(\frac{1}{2})^i)^n = (\frac{1}{2})^n + ((\frac{1}{2})(\frac{1}{2})^1)^n + ((\frac{1}{2})(\frac{1}{2})^2)^n + ((\frac{1}{2})(\frac{1}{2})^3)^n + ... + ((\frac{1}{2})(\frac{1}{2})^k)^n$
Distribute the exponent $n$ to get as a constant $\left(\frac{1}{2}\right)^n$ You get $$\sum_{i=0}^k \left(\frac{1}{2}\right)^n \left(\frac{1}{2^n}\right)^i$$ can you figure what is $r$ and what is $a$?