Geodesic
Monotonicity of Certain Riemann-type Sums
The Teaching of Mathematics, XV (2012) no. 2, p. 113

Voir la notice de l'article provenant de la source eLibrary of Mathematical Institute of the Serbian Academy of Sciences and Arts

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).
Classification : 1AMS97I30 2MESCI35
Keywords: Monotone sequence, Riemann sums. 
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     author = {Szil\'ard Andr\'as},
     title = {Monotonicity of {Certain} {Riemann-type} {Sums}},
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Szilárd András. Monotonicity of Certain Riemann-type Sums. The Teaching of Mathematics, XV (2012) no. 2, p. 113 . http://geodesic.mathdoc.fr/item/TM2_2012_XV_2_a2/