Question

1/3.1!+1/4.2!+1/5.3!.....infinity=
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Answer 1

Given :  1/3.1!+1/4.2!+1/5.3! + .................+∞  

To find : Sum of given infinite series

Solution:

1/3.1!+1/4.2!+1/5.3! + .................+∞

= 2/2.3.1!   +   3/3.4.2!   + 4/4.5.3! + .................+∞

= 2/3.2.1!   +   3/4.3.2!   + 4/5.4.3! + .................+∞

= 2/3!   +   3/4!   + 4/5! +.................+ ∞

= ( 3 - 1)3!  + (4 - 1)4!  +  (5 - 1)5! +........................+∞

= (3/3!  + 4/4! + 5/5! +..................+∞ ) - ( 1/3! + 1/4!  + 1/5! +.....................+∞ )

= (3/3.2!  + 4/4.3! + 5/5.4! +..................+∞ ) - ( 1/3! + 1/4!  + 1/5! +.....................+∞ )

= (1/ 2!  + 1/ 3! + 1/ 4! +..................+∞ ) - ( 1/3! + 1/4!  + 1/5! +.....................+∞ )

=  1/ 2! + (1/ 3! + 1/ 4! +..................+∞ ) - ( 1/3! + 1/4!  + 1/5! +.....................+∞ )

= 1/2!

= 1/2

1/3.1!+1/4.2!+1/5.3! + .................+∞  = 1/2

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