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Find the value of $$\sum\limits_{n=1}^{\infty} \frac{2^{n}}{x^{2^n}+1}$$

I recently came across a question in which we had to find the value of the above question. The question seemed simple at first glance but the term $2^n$ in the numerator is posing me with a problem. I could neither reduce it to a telescopic series nor could I compare it with any standard expansion. I've run out of ideas. Would someone please help me to solve this problem?

Thanks for help.

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Hint: Multiply by $x^{2^n}-1$ in the numerator and denominator.

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  • $\begingroup$ Thank you Shashwat1337 $\endgroup$ – AK2021 Oct 5 at 6:14
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The series is telescopic: note that $$\frac{2^{n}}{x^{2^n}+1}=\dfrac{2^n}{x^{2^{n}}-1} - \dfrac{2^{n+1}}{x^{2^{n+1}}-1}.$$

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