Friday, March 1, 2019

calculus - Convergence test from Demidovich: $sum_{n=1}^{infty} frac{n^{n + frac{1}{n}}}{(n + frac{1}{n})^n}$


I'm just learning how to test series for convergence and have encountered this series from the Demidovich's book and I can't really decide what criteria should I use. Could you please give me some hint(s)?


$$\sum_{n=1}^{\infty} \frac{n^{n + \frac{1}{n}}}{(n + \frac{1}{n})^n}$$



Answer



Hint: Since $n^{1/n} \to 1,$ you can forget about it. We're left with an $n$th term equal to


$$\frac{n^n}{(n+1/n)^n} = \left ( \frac{1}{1 + 1/n^2} \right )^n.$$


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