1. Tardigrade
  2. Question
  3. Mathematics
  4. Let S= displaystyle∑k=0∞ (2k/72k+1) then (1/S) is equal to

Question Error Report

Thank you for reporting, we will resolve it shortly

Back to Question

Q. Let $S=\displaystyle\sum_{k=0}^{\infty} \frac{2^{k}}{7^{2^{k}}+1}$ then $\frac{1}{S}$ is equal to

Sequences and Series

Solution:

$S=\displaystyle\sum_{k=0}^{\infty}\left(\frac{2^{k}}{7^{2^{k}}+1}\right)$
$S=\frac{2^{0}}{7^{1}+1}+\frac{2^{1}}{7^{2}+1}+\frac{2^{2}}{7^{4}+1}+\frac{2^{3}}{7^{8}+1}$
$\left(\frac{1}{7-1}-\frac{2}{7^{2}-1}\right)+\left(\frac{2}{7^{2}-1}-\frac{2^{2}}{7^{4}-1}\right)+\left(\frac{2^{2}}{7^{4}-1}-\frac{2^{3}}{7^{8}-1}\right)+\ldots$
$S=\frac{1}{6} \Rightarrow \frac{1}{5}=6$