Let $f \in [0,1] \to \mathbb{R}$ be 2 times differentiable function, with continuous second derivative. Compute $\lim_{n \to \infty} n(n( \int_{0}^{1} f(x) \ dx - \frac{1}{n}\sum_{k=1}^{n} f(\frac{k}{n}))-\frac{f(0)-f(1)}{2} )$ .
The first thing I noticed was that the limit terms in the big bracket is $0$, as you can see from this idea in this link Limit of a Riemann Sum and Integral , with $\lim_{n \to \infty}n( \int_{0}^{1} f(x) \ dx - \frac{1}{n}\sum_{k=1}^{n} f(\frac{k}{n}))=\frac{f(0)-f(1)}{2}$, but further I don't know how to continue.If someone could help me I would be grateful.
