Manipulate the series and find limit.

Question

Let $$ L= \lim_{n\to \infty} \sum_{r=0}^n \frac{2^r}{5^{2^r}+1},$$ then find $L$.

I tried various ways to manipulate it to difference series, but failed to do so. Please help

Thankyou

Answer 1

Hint:

$$\dfrac1{5^{2^r}-1}-\dfrac1{5^{2^r}+1}=\dfrac2{\left(5^{2^r}\right)^2-1^2}=\dfrac2{5^{2^{r+1}}-1}$$

$$\implies\dfrac{2^r}{5^{2^r}-1}-\dfrac{2^{r+1}}{5^{2^{r+1}}-1}=\dfrac{2^r}{5^{2^r}+1}$$

Put $r=0,1,2,\cdots,n-1,n$ to recognize the Telescoping pattern.