Saturday, September 9, 2017

calculus - Sum of the series:frac11cdot2+frac1cdot31cdot2cdot3cdot4+cdots



I have a question where i couldn't find any clue.
The question is




112+131234+135123456+



I could get the general term as tn=1357(2n1)1234562n



I have also tried it to form the sequence in the telescopic form.But couldn't get. Any hint will be appreciated.


Answer



tn=1357(2n1)1234562n=(2n1)!!(2n1)!!(2n)!!=1(2n)!!=12nn!



Therefore this sum does not go to infinity.




Invoking ex=n=0xnn!, we can compute the sum: n=1(12)nn!=n=0(12)nn!1=e121


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