I, along with my friend was studying for real analysis yesterday and we came across this question:
Let $p > 0$, $q > 1$, $m > k$. Evaluate lim of sequence $s_n$ when $n$ goes to infinity given
$$s_n = \sum_{k = 1}^{n}\frac{k^p}{n^{p+1}}$$
My friend said that since this is Riemann Sum of integral of $x^p$ with limits for $0$ to $1$, we can simply integrate $x^p$ and get the answer. I am really confused as to how he made the link with Riemann sum. If you can please clarify on the explanation I will be really grateful.
hint: $\displaystyle \lim_{n\to \infty}s_n = \displaystyle \lim_{n\to \infty}\dfrac{1}{n}\displaystyle \sum_{k=1}^n \left(\dfrac{k}{n}\right)^p = \displaystyle \int_{0}^1 x^pdx=.....$. To see the details: Set $f(x) = x^p, a=0,b=1, x_i = \dfrac{i}{n}$. Thus $\triangle x_i = \dfrac{1}{n}$. Thus: $s_n = \displaystyle \sum_{k=1}^n f(x_k)\triangle x_k= \dfrac{1}{n}\displaystyle \sum_{k=1}^n \left(\dfrac{k}{n}\right)^p$