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Question: The sum of the series \(\frac{1}{2!} - \frac{1}{3!} + \frac{1}{4!} - .....\) is....

The sum of the series 12!−13!+14!−.....\frac{1}{2!} - \frac{1}{3!} + \frac{1}{4!} - ..... is.

A

e

B

e−12e^{- \frac{1}{2}}

C

e−2e^{- 2}

D

None of these

Answer

None of these

Explanation

Solution

We know,

11.2−12.3+13.4−14.5+.....∞=\frac{1}{1.2} - \frac{1}{2.3} + \frac{1}{3.4} - \frac{1}{4.5} + .....\infty =

Put x=−1x = - 1 log⁡ee4\log_{e}\frac{e}{4}

⇒ log⁡e4\log_{e}4