Here is a question on limits. I would like to ask help. Here it goes: $$\lim_{N\to\infty}\left(\frac{\sum_{j=0}^{N}\left(\frac{j}{N}\right)^{n+1}}{\sum_{j=0}^{N}\left(\frac{j}{N}\right)^{n}}\right)$$ I do not know where to start but so far, I think I have to use L'Hopital's Rule, maybe because it becomes $\frac{0}{0}$, which is indeterminate? I hope somebody can explain. I really can't figure out. Thanks.
P.S. Some texts simplified this as $$\frac{n+1}{n+2}$$Is this correct? Why?
Each of the sums in your limit is a Riemann sum. Multiply up and down in the fraction by $1/N$, and use, e.g.,
$$\lim_{N \to \infty} \frac{1}{N}\sum_{j=0}^N \left (\frac{j}{N}\right )^n = \int_0^1 dx \, x^n = \frac{1}{n+1}$$