1. ## A Limit

Let $\displaystyle I_n(p)=\frac{1}{n^{p+1}}\sum_{j=1}^n j^p, \ \ p \in \mathbb{R}.$ We know that if $\displaystyle p > -1,$ then $\displaystyle \lim_{n\to\infty}I_n(p)=\int_0^1 x^p \ dx = \frac{1}{p+1}.$

Question 1: Suppose $\displaystyle p \in \mathbb{N}.$ Evaluate $\displaystyle \lim_{n\to\infty} n \left(I_n(p) - \frac{1}{p+1} \right).$

Question 2: (maybe harder!) What is the above limit for $\displaystyle -1 < p \in \mathbb{R}$ ?

2. For Q.1 the answer 1/2 comes straight out of Faulhaber's formula. But that is independent of p, which makes it very tempting to think that the answer to Q.2 might be the same.

3. Originally Posted by Opalg

... which makes it very tempting to think that the answer to Q.2 might be the same.
but (in Q.2) at p = 0 the limit is 0 not 1/2.

4. Originally Posted by NonCommAlg
but (in Q.2) at p = 0 the limit is 0 not 1/2.
Hmmm yes, that's right. And if p<0 then the Riemann sum underestimates the integral, so the limit is going to be negative.

I'll stop making conjectures at that point.