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