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The sum of the series- 1/(x+1) + 2/(x²+1) + 2²/(x⁴+1) + ..... + 2^100/(x^100+1) when x = 2 is :

1. 1 + 2^101/(4^101 − 1)
2. 1 + 2^100/(4^101 − 1)
3. 1 − 2^100/(4^100 − 1)
4. 1 − 2^101/(4^101 − 1)

Correct answer: 1 − 2^101/(4^101 − 1)

Solution

The correct option is derived from evaluating the series at x = 2, where each term simplifies and the series converges to a specific value. The manipulation of the series leads to the conclusion that the sum can be expressed as 1 minus a fraction involving powers of 2 and 4, which matches the given correct option.

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