limits - Summation of an infinite Exponential series

Q. Find the value of -

$$\lim_{n\to\infty} \sum_{r=0}^{n} \frac{2^r}{5^{2^r} +1}$$

My attempt - I seem to be clueless to this problem. Though I think that later terms would be much small and negligible( Imagine how large would be $5^{2^r}$ after 3-4 terms), so I calculated the sum of first 3-4 terms and my answer was just around the actual answer ( just a difference of $0.002$ ). But I wonder if there is an actual method to solve this problem? If it is, would you please share it to me?

Any help would be appreciated