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JEE Main202126 Feb 2021Evening ShiftMathematicsSequences and SeriesActual

The sum of the series ∑ n = 1 ∞ n 2 + 6 n + 10 2 n + 1 ! is equal to

Options

  1. A41 8 e + 19 8 e - 1 + 10
  2. B41 8 e + 19 8 e - 1 - 10
  3. C- 41 8 e + 19 8 e - 1 - 10
  4. D41 8 e - 19 8 e - 1 - 10

Correct answer

D. 41 8 e - 19 8 e - 1 - 10

Step-by-step solution

T n = n 2 + 6 n + 10 2 n + 1 ! = 4 n 2 + 24 n + 40 4 . 2 n + 1 ! = 2 n + 1 2 + 20 n + 39 4 . 2 n + 1 ! = 2 n + 1 2 + 2 n + 1 . 10 + 29 4 2 n + 1 ! = 1 4 2 n + 1 2 2 n + 1 2 n ! + 2 n + 1 10 2 n + 1 2 n ! + 29 2 n + 1 ! = 1 4 2 n + 1 2 n ! + 10 2 n ! + 29 2 n + 1 ! = 1 4 1 2 n - 1 ! + 11 2 n ! + 29 2 n + 1 ! S 1 = 1 1 ! + 1 3 ! + 1 5 ! + … = e - 1 e 2 S 2 = 11 1 2 ! + 1 4 ! + 1 6 ! + … = 11 e + 1 e - 2 2 S 3 = 29 1 3 ! + 1 5 ! + 1 7 ! + … = 29 e - 1 e - 2 2 Now, S = 1 4 S 1 + S 2 + S 3 = 1 4 e 2 -

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