I'm having a little trouble with a question that requires me to interpret a limit as a Riemann sum for an integral. However, I'm having trouble identifying which aspects of the limit correspond to the parts of the function I need to know to relate it to an integral.

The limit is as follows: $\lim_{n\to\infty}\frac{(\sqrt[n]e)+(\sqrt[n]e^2)+...+(\sqrt[n]e^{2n})}{n}$

I'm not entirely sure how to proceed. I know that $1/n=Δx$ but I don't know which function the integral is representing.

  • $\begingroup$ Are you sure your individual terms are correct? Can you add a couple more terms near the beginning? It's difficult to determine the pattern. Are the exponents of $e$ increasing by $1$ with each successive term? $\endgroup$ – MPW Apr 17 at 16:06
  • $\begingroup$ I'm sure, I've taken it from an old test so unless there was an error in the way that it was printed, this is how the question was intended $\endgroup$ – LucasLyons Apr 17 at 16:25
  • $\begingroup$ I dont think there is an error. In my answer I showed how it can be written as a Riemann sum. $\endgroup$ – Mark Apr 17 at 16:33
  • $\begingroup$ @Mark : It’s just not clear how many terms there are because the pattern is ambiguous with so few terms $\endgroup$ – MPW Apr 17 at 17:57

Hint: the limit can be written as $\lim_{n\to\infty}2\sum_{k=1}^{2n} e^{2\frac{k}{2n}}\frac{1}{2n}$. Is it more clear now?


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