The sum of the series [1 x + 1 + 2 x^2 + 1 + 2^2 x^4 + 1 + ......

Question

The sum of the series

$${1 \over {x + 1}} + {2 \over {{x^2} + 1}} + {{{2^2}} \over {{x^4} + 1}} + ...... + {{{2^{100}}} \over {{x^{{2^{100}}}} + 1}}$$ when x = 2 is :

Accepted Answer

$$S = {1 \over {x + 1}} + {2 \over {{x^2} + 1}} + {{{2^2}} \over {{x^4} + 1}} + ...... + {{{2^{100}}} \over {{x^{{2^{100}}}} + 1}}$$

$$S + {1 \over {1 - x}} = {1 \over {1 - x}} + {1 \over {x + 1}} + ...... = {2 \over {1 - {x^2}}} + {2 \over {1 + {x^2}}} + ....$$

$$S + {1 \over {1 - x}} = {{{2^{101}}} \over {1 - {x^{400}}}}$$

$$S = 1 - {{{2^{101}}} \over {{2^{400}} - 1}}$$

Solutions · 1