The given problem is a problem of calculus. I have tried this problem only to find that the numerator is a divergent series. No idea how to do this problem
$$\lim_{n\to\infty}\frac{1+2^{1/2}+3^{1/3}+\cdots+n^{1/n}}{n}$$
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Please remember that you can choose an answer among the given if the OP is solved, more details here https://meta.stackexchange.com/questions/5234/how-does-accepting-an-answer-work – user Apr 19 '18 at 19:34
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HINT
By Stolz-Cesaro
$$\lim_{n \to \infty} \frac{\sum_{k=1}^n k^\frac1k}n=\lim_{n\to\infty}\quad (n+1)^{\frac1{n+1}}=1$$
user
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You can prove in general that if $a_n\to a$ as $n\to \infty$, then $$ \sum_{i = 1}^n\frac{a_i}{n}\to a $$ as well. Now note that $\sqrt[n]{n}\to 1$, and you're done.
Arthur
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