Friday, 21 February 2014

calculus - Convergence of sumlimitsin=1nftyleft(efrac1n1frac1nright)


I've got in my assignment to show if the following series converges or diverges.
n=1(e1n11n)





Attempt:
n=1(e1n11n)=n=1(1+1n+12!n2+13!n3+...11n)=n=1(12!n2+13!n3+...)=n=1(1n+12!n2+13!n3+...)



At this point I'm lost. I tried using D'Alambert as follows:




lim



which I tried to simplify with basic limit laws (hopefully correctly):



\lim\frac{\sqrt[n+1]{e}-1}{\sqrt[n]{e}-1} = 1



I don't know where to go from here. Thank you for all your help in advance.

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