# Limit as a definite integral (Riemann Sum)

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.

• 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? – MPW Apr 17 at 16:06
• 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 – LucasLyons Apr 17 at 16:25
• I dont think there is an error. In my answer I showed how it can be written as a Riemann sum. – Mark Apr 17 at 16:33
• @Mark : It’s just not clear how many terms there are because the pattern is ambiguous with so few terms – MPW Apr 17 at 17:57

## 1 Answer

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?