Find the sum : $\displaystyle\sum_{i=0}^n \frac{2^i}{1+x^{2^{i}}}$

Question

$\displaystyle\sum_{i=0}^n \frac{2^i}{1+x^{2^{i}}}$

What technique is applicable here? I can't find a way to manipulate this sum to make it telescope. Just guide me.

Answer 1

$$\frac{2^{i+1}}{1-x^{2^{i+1}}} = \frac{2^{i+1}}{(1-x^{2^i})(1+x^{2^i})} = 2^i \left ( \frac1{1-x^{2^i}} + \frac1{1+x^{2^i}}\right )$$

Therefore

$$\frac{2^i}{1+x^{2^i}} = \frac{2^{i+1}}{1-x^{2^{i+1}}} - \frac{2^i}{1-x^{2^i}}$$

Can you take it from here?