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 :
Correct Option: , 4
Solution:
$S=\frac{1}{x+1}+\frac{2}{x^{2}+1}+\frac{2^{2}}{x^{4}+1}+\ldots \frac{2^{100}}{x^{2^{100}}+1}$
$S+\frac{1}{1-x}=\frac{1}{1-x}+\frac{1}{x+1}+\ldots .=\frac{2}{1-x^{2}}+\frac{2}{1+x^{2}}+\ldots$
$S+\frac{1}{1-x}=\frac{2^{101}}{1-x^{2^{101}}}$
Put $x=2$
$S=1-\frac{2^{101}}{2^{2^{101}}-1}$
Not in option (BONUS)
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