Monotonicity of Certain Riemann-type Sums

Szilárd András

In this short note we prove with elementary techniques that the sequence $x_n=\sum_{k=1}^n\frac{n}{n^2+k^2}$ is increasing and its limit is $\frac{\pi}{4}$. Moreover, we give a sufficient condition for the monotonicity of some Riemann-type sums assigned to uniform subdivisions as a function of the number of the intervals from the subdivision. This mathematical content came up in a group discussion during an IBL centered teacher training activity and reflects a crucial problem is implementing IBL teaching attitudes in the framework of a highly scientific curricula (such as the Romanian mathematics curricula for upper secondary school).