Q:

Convergence of the series $\sum_{n=0}^{\infty} \frac{5^{2n-1}}{10^{2n -1}}$.

I am trying to find out the sum (I just derived these from 2 + 0.5 + 0.125 + 0.03125 + ...):
$$\sum_{n=0}^{\infty} \frac{5^{2n-1}}{10^{2n -1}}$$
It's confusing me because it doesn't match $${ar}^{n-1}$$ the power by which $r$ is raised.  

A:

$\frac{a^k}{b^k} = (\frac{a}{b})^k$. Also $a^{2k} = (a^2)^k$, and $a^{2k-1} = \frac{1}{a} \cdot a^{2k}$. This should allow you to simplify it.