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
limn→∞xn+1nn(n+1)n+1xn<1
limn→∞xnn(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
limn→∞xnn(n+1)n+1<1.
But
limn→∞xnn(n+1)n+1=limn→∞x(1+1n)n(n+1).
As n→∞, the limit limn→∞(1+1n)n=e. Thus the limit
limn→∞x(1+1n)n(n+1)=0∀x.
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