Question

The sum of the series $\frac{1}{x+1}+\frac{2}{x^2+1}+\frac{2^2}{x^4+1}+\dots ..+\frac{2^{100}}{x^{2^{100}}+1}$ when $x=2$ is:

• A. $1+\frac{2^{101}}{4^{101}-1}$
• B. $1+\frac{2^{100}}{4^{101}-1}$
• C. $1-\frac{2^{100}}{4^{100}-1}$
• D. $1-\frac{2^{101}}{2^{101}-1}$

Answer 1

Answer: (D)

$S=\frac{1}{x+1}+\frac{2}{x^2+1}+\frac{2^2}{x^4+1}+\dots \frac{2^{100}}{x^{2^{100}}+1}$
$S+\frac{1}{1-x}=\frac{1}{1-x}+\frac{1}{x+1}+\dots =\frac{2}{1-x^2}+\frac{2}{1+x^2}+\dots$
$S+\frac{1}{1-x}=\frac{2^{101}}{1-x^{2^{101}}}$
Put $x=2$
$S=1-\frac{2^{101}}{2^{2^{101}}-1}$