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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 :

JEE · Math · previous-year question

  1. A.$$1 + {{{2^{101}}} \over {{4^{101}} - 1}}$$
  2. B.$$1 + {{{2^{100}}} \over {{4^{101}} - 1}}$$
  3. C.$$1 - {{{2^{100}}} \over {{4^{100}} - 1}}$$
  4. D.$$1 - {{{2^{101}}} \over {{2^{400}} - 1}}$$correct

Answer

D. $$1 - {{{2^{101}}} \over {{2^{400}} - 1}}$$

Explanation

$$S = {1 \over {x + 1}} + {2 \over {{x^2} + 1}} + {{{2^2}} \over {{x^4} + 1}} + ...... + {{{2^{100}}} \over {{x^{{2^{100}}}} + 1}}$$ $$S + {1 \over {1 - x}} = {1 \over {1 - x}} + {1 \over {x + 1}} + ...... = {2 \over {1 - {x^2}}} + {2 \over {1 + {x^2}}} + ....$$ $$S + {1 \over {1 - x}} = {{{2^{101}}} \over {1 - {x^{400}}}}$$ $$S = 1 - {{{2^{101}}} \over {{2^{400}} - 1}}$$

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