Question 33
Let $f : [0, 1] \to \mathbb{R}$ be a derivable function, with continuous derivative, and let $s_n = \sum_{k=1}^{n} f\left(\frac{k}{n}\right)$.
Prove that the sequence $(s_{n+1} - s_n)_{n \in \mathbb{N}^*}$ converges to $\int_0^1 f(x) dx$.
Prove that the sequence $(s_{n+1} - s_n)_{n \in \mathbb{N}^*}$ converges to $\int_0^1 f(x) dx$.