How can we solve the following?:
$$\lim_{N\to \infty} \sum_{r=1}^N \frac{2^r}{(1+5^{2^r})} $$
The answer according to Wolfram Alpha is $\frac{1}{12}$, but I have no idea how it can be calculated on paper without summing it infinitely (which apparently is what Wolfram Alpha does).
Hint:
$$\dfrac{2^r}{5^{2^r}-1}-\dfrac{2^r}{5^{2^r}+1}=\dfrac{2^{r+1}}{5^{2^{r+1}}-1}$$
Use Telescoping series