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JEE Main202126 Feb 2021Evening ShiftMathematicsSequences and SeriesActual

If 0 < a , b < 1 , and tan - 1 a + tan - 1 b = π 4 , then the value of a + b - a 2 + b 2 2 + a 3 + b 3 3 - a 4 + b 4 4 + … is :

Options

  1. Alog e e 2
  2. Be
  3. Ce 2 - 1
  4. Dlog e 2

Correct answer

D. log e 2

Step-by-step solution

tan - 1 a + tan - 1   b = π 4   0 < a , b < 1 ⇒ a + b 1 - a b = 1 a + b = 1 - a b a + 1 b + 1 = 2 Now a - a 2 2 + a 3 3 + … + b - b 2 2 + b 3 3 + … = log e 1 + a + log e 1 + b ( ∵ expansion of log e 1 + x ) = log e 1 + a 1 + b = log e 2

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