Thursday, 5 September 2013

sequences and series - How to solve limnto+inftyn2left(eleft(1+frac1nright)nright)?




I am trying to solve this sequence limit: limn+n2(e(1+1n)n)

but the only elementary way I found to solve it is to prove that $$ \left(1+\frac1n\right)^n+\frac1n for n>3 and so n2(e(1+1n)n)>n21n=1n+
.



However proving the first inequality is not straightforward so I would like to know if there exist a more direct way to solve it.


Answer



I used the fact : limnn(e(1+1n)n)=e2



For the large n,



n2(e(1+1n)n)e2nwheren


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