6
$\begingroup$

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.

| cite | improve this question | |
$\endgroup$
5
$\begingroup$

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

| cite | improve this answer | |
$\endgroup$
  • 1
    $\begingroup$ Thank you Shashwat1337 $\endgroup$ – AK2021 Oct 5 '19 at 6:14
3
$\begingroup$

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

| cite | improve this answer | |
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.