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Question

13!+25!+37!+....is equal to


A

e

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B

e-12

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C

e4

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D

e6

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Solution

The correct option is B

e-12


Explanation for the correct option:

Step 1. Find the sum of the given series:

Given 13!+25!+37!+.

=01!+13!+25!+37!+.

S=n=0n(2n+1)!

Multiply and divide both numerator and denominator by 2, also add and subtract 1 from numerator

S=12n=0(2n+1-1)(2n+1)!

=12n=0(2n+1)(2n+1)!n=01(2n+1)!

=12n=012n!n=o1(2n+1)!

We know, e=1+11!+12!+13!+....+ ..….(i)

e-1=1-11!+12!13!+....+ ……(ii)

Step 2. Add equation (i) and (ii) and Subtract equation (iii) from equation (i):

(e+e-1)=2n=01(2n)! ……(iii)

and (e-e-1)=2n=01(2n+1)! ……(iv)

Step 3. Substitute equation (iii) and (iv) in S:

S=12n=012n!n=01(2n+1)!

=12(e+e-1)2(e-e-1)2

=e-12

Hence, option ‘B’ is Correct.


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