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

Question

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.

Is my editing correct? " but the term $n^2$ in the numerator" Do you mean $2^n$? — Robert Z, Oct 05 '19 at 06:14
Possible duplicate of Find the Sum of the Series: $1/(x+1) + 2/(x^2 + 1) + 4/(x^4 +1) +\cdots$ $n$ terms — metamorphy, Oct 05 '19 at 06:35

score 6 · Accepted Answer · answered Oct 05 '19 at 06:11

6

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

answered Oct 05 '19 at 06:11

shsh23

1,135

1

Thank you Shashwat1337 – AK2021 Oct 05 '19 at 06:14

score 4 · Answer 2 · answered Oct 05 '19 at 06:25

4

The series is telescopic: note that $$\frac{2^{n}}{x^{2^n}+1}=\dfrac{2^n}{x^{2^{n}}-1} - \dfrac{2^{n+1}}{x^{2^{n+1}}-1}.$$

answered Oct 05 '19 at 06:25

Robert Z

145,942

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

2 Answers2