Application of a Mean Value Theorem for Integrals to Series Summation

A Short and Elementary Proof of the Basel Problem

“…3. Our proof is closely connected with the ones in [25] and [73]. We came across these two references shortly after a first submission to a journal in December 2014.…”

“…We came across these two references shortly after a first submission to a journal in December 2014. To tell the truth, our proof is independent of [25] and [73]. Moreover, it is the result of a marvelous serendipity when we were preparing for a class on integration methods.…”

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By doing a slight change to a beautiful and widely unknown argument by E. L. Stark [25] we get a candidate to be considered as one of the shortest and most elementary proofs of the celebrated Basel Problem. Furthermore, we give a comprehensive list of references on this topic, displayed in chronological order from Euler to present.

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On the discretization of probability density functions and the continuous Rényi entropy

“…The starting point is the SMVT for integrals [9][10][11][12][13], i.e., if f (x) is strictly monotonic in the interval (p, q), where p < q, and if ρ(x) ≥ 0 is integrable over that interval, then there exists at least one point ξ (with p ≤ ξ ≤ q) such that…”