Question

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 ______.

Options

• `1 - 2^101/(4^101 - 1)`

• `1 + 2^101/(4^101 - 1)`

• `1 - 2^100/(4^100 - 1)`

• `1 + 2^101/(4^101 - 1)`

Answer 1

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 `underlinebb(1 - 2^101/(4^101 - 1))`.

Explanation:

Adding and subtracting `1/(1 - x)`

∴ S = `1/(1 - x) - 1/(1 - x) + 1/(x + 1) + 2/(x^2 + 1) + 2^2/(x^4 + 1) + .... + 2^100/(x^(2^100 + 1)`

∴ S = `1/(x - 1) + 2/(1 - x^2) + 2/(x^2 + 1) + 2^2/(x^4 + 1) + .... + 2^100/(x^(2^100) + 1)`

∴ S = `1/(x - 1) - 2^101/(x^(2^101) - 1)`

For x = 2 ⇒ S = `1 - 2^101/(2^(2^101) - 1)` = `1 - 2^101/(4^101 - 1)`