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1+(1)/(3!)+(1)/(5!)+....=...

`1+(1)/(3!)+(1)/(5!)+....=`

A

e

B

`(e)/(2)`

C

`(1)/(2)(e+e^(-1))`

D

`(1)/(2)(e-e^(-1))`

Text Solution

Verified by Experts

The correct Answer is:
D
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(1)/(2!)-(1)/(3!)+(1)/(4!)-(1)/(5!)+....=

(1)/(1.3)+(1)/(2)((1)/(3.5))+(1)/(3)((1)/(5.7))+....=

(1)/(1.2)+(1)/(3.4)+(1)/(5.6)+....

I:(1)/(1.2)+(1)/(3.4)+(1)/(5.6)+....log_(e)2 II:(1)/(1.2)-(1)/(2.3)+(1)/(3.4)-(1)/(4.5)+....=2log_(e)2-1

(1)/(1.3) + (1)/(3.5) + (1)/(5.7) + …. (n-3) terms

underset(n to oo)lim {(1)/(1.3)+(1)/(3.5)+(1)/(5.7)+.....+(1)/((2n-1)(2n+1))}=

Prove by the method of induction, (1)/( 1.3) + (1)/( 3.5) + (1)/( 5.7) + . . . + (1)/( (2n - 1)(2n + 1)) = (n)/(2 n +1)