Thursday, 11 February 2016

real analysis - For what values of x does the series converge: sumlimitsinftyn=1fracxnnn?




For what values of x do the following series converge or diverge



n=1xnnn



I tried to solve this using the ratio test where the series converge when



limnxn+1nn(n+1)n+1xn<1



limnxnn(n+1)n+1<1




but then I am not sure what to do next.



Please give me some ideas or hints on how to solve this question, thanks to anybody who helps.


Answer



Staring from the last step you have shown we have
limnxnn(n+1)n+1<1.


But
limnxnn(n+1)n+1=limnx(1+1n)n(n+1).

As n, the limit limn(1+1n)n=e. Thus the limit
limnx(1+1n)n(n+1)=0x.



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