Question

The sum of the series $\frac{1}{x+1}+\frac{2}{x^{2}+1}+\frac{2^{2}}{x^{4}+1}+\ldots . .+\frac{2^{100}}{x^{2^{100}}+1}$ 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)

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