limit of sum $\dfrac{(-1)^{n-1}}{2^{2n-1}}$ What is: $$\sum^{\infty}_{n=1}\dfrac{(-1)^{n-1}}{2^{2n-1}}$$
I have done a Leibniz convergence test and proved that this series converges, but I do not know how to find the limit. Any suggestions?

We can appeal to the sum of a geometric series as follows:
$$\begin{align}
\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{2^{2n-1}}&=\sum_{n=0}^{\infty}\frac{(-1)^{n}}{2^{2n+1}}\\\\
&=\frac12\sum_{n=0}^{\infty}\left(\frac{-1}{4}\right)^n\\\\
&=\frac12\frac{1}{1+\frac14}\\\\
&=\frac25
\end{align}$$
