The sum of the series 1 x + 1 + 2 x 2 + 1 + 2 2 x 4 + 1 + … + 2 100 x 2 100 + 1 when x = 2 is:

Adding and subtracting $\frac{1}{x-1}$

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

$S=\frac{1}{x-1}+\left(\frac{1}{x+1}-\frac{1}{x-1}\right)+\frac{2}{x^2+1}+\frac{2^2}{x^4+1}+\dots +\frac{2^{100}}{x^{2^{100}}+1}$

$S=\frac{1}{x-1}+\left(\frac{2}{x^2+1}+\frac{-2}{x^2-1}\right)+\frac{2^2}{x^4+1}+\dots .+\frac{2^{100}}{x^{2^{100}}+1}$

$S=\frac{1}{x-1}+\left(\frac{-2^2}{x^4-1}\right)+\frac{2^2}{x^4+1}+\dots .+\frac{2^{100}}{x^{2^{100}}+1}$

$S=\frac{1}{x-1}-\frac{2^{101}}{x^{2^{101}}-1}$

For $x=2\Rightarrow S=1-\frac{2^{101}}{2^{2^{101}}-1}=1-\frac{2^{101}}{4^{101}-1}$