TITLE: When do Riemann sums converge monotonically?
QUESTION [3 upvotes]: Let $f : [0, 1] \to \mathbb{R}$ be a 'well behaved' function and define $c_n$ by the Riemann sums
$$
c_n := \frac{1}{n} \sum_{k = 1}^n f\left(\frac{k - \theta}{n}\right)
$$
for some given $\theta \in [0, 1]$. Then obviously $c_n$ will converge to the integral of $f$. Under which conditions on $f$ will this convergence be monotone? In particular, will the convergence be monotone if $f$ is twice differentiable with $f > 0$, $f' < 0$ and $f'' > 0$?

REPLY [0 votes]: Well, if $f$ is decreasing and convex, it is true that $c_n$ increases monotonically, if $\theta<0$.
The proof is fairly straightforward, left as an exercise to the reader.
See the question on MO.