Geodesic
Monotonicity of Certain Riemann-type Sums
The Teaching of Mathematics, XV (2012) no. 2, p. 113 Cet article a éte moissonné depuis la source eLibrary of Mathematical Institute of the Serbian Academy of Sciences and Arts

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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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     number = {2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TM2_2012_XV_2_a2/}
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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/