Basel Problem Research Articles

On the Basel Problem and the Square of Gregory’s Series

Summary If we consider the Madhava–Gregory–Leibniz series π 4 = 1 − 1 3 + 1 5 − ⋯ , which is commonly referred to as the Gregory series, and then compare it to Euler’s formula π 2 8 = 1 + 1 3 2 + 1 5 2 + ⋯ , the following question arises: Can this latter formula be derived by squaring both sides of the former? There have been several proofs of Euler’s formula, or its equivalent formulation ζ ( 2 ) = π 2 / 6 , based on the idea of squaring 1 − 1 3 + 1 5 − ⋯ = π 4 , including a proof presented in a letter from Euler to Goldbach dating from 1742. We consider the history of proofs of this form, and we offer another simple proof of ζ ( 2 ) = π 2 / 6 that also relies on squaring Gregory’s series.

Solution of the Basel Problem in the Framework of Distribution Theory

A simple proof of Euler’s formula which states that the sum of the reciprocals of all natural numbers squared equals ζ(2)=π2/6\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\zeta (2)=\\pi ^2/6$$\\end{document} is presented based on the distribution theory introduced by Laurent Schwartz. Additional identities are obtained as a byproduct of the derivation.

The Subseries are Dense in the Basel Problem

"The Subseries are Dense in the Basel Problem." The American Mathematical Monthly, ahead-of-print(ahead-of-print), p. 1

Beyond the Basel Problem, Part I

Summary In the 18th century, the problem of finding the sum of the reciprocals of the squares was known as the Basel problem. It was first solved by Euler, who, in 1735, showed the sum evaluated to . We go beyond the Basel problem by evaluating the sum of where , for some integers b and a. In particular, we examine the special cases and .

Combining Euler series and series from continuations of the Bernoulli scheme

This article describes a new approach to constructing numerical series which converge to an irrational number, a two series combination being proposed. The first series is constructed from the probabilistic interpretation of summing numerical series, which is the authors’ continuation of the Bernoulli scheme. Our ideas are based on the rejection of finiteness and independence of trails. A probabilistic urn model allows us to get a convergent numerical series as well as a series coinciding with the elements of Leibniz harmonic triangle diagonals. The second series used in the considered approach is presented by different Euler series which are extensions of the Basel problem. The combination of two numerical series makes it possible to sum unknown numerical series in closed form. A new reading of the Leibniz harmonic triangle and Euler series variations as well as an approach to finding probabilistic urn models of the summing series are proposed. It is symbolic that the paper considers the ways of connecting the “Basel series” and the series obtained from the continuation of the Bernoulli scheme taking into account that Jacob and Johann Bernoulli worked at the University of Basel. The adoption of the method of combining two numerical series enables to obtain results which have not been previously published in scientific literature.

Theory and applications on a new generalized Laplace‐type integral transform

In this paper, several theorems and relations are examined by using a generalized Laplace‐type integral transform. A generalization of the harmonic oscillator in a non‐resisting and resisting medium problems, initial‐boundary problems, and integral equations is solved via this integral transform. Furthermore, the well‐known series entitled as Basel problem is obtained in a similar way. Moreover, we compare numerically classical and newly introduced by us integral transform.

Basel Problem: Historical perspective and further proofs from stochastic processes

In this note, we offer a historical perspective on solutions of the Basel problem. In particular, we have a closer look at some of the less famous results by Euler E41 and provide a review of a selection of the assemblage of earlier proofs. Moreover, we show how to generate further proofs using Karhunen-Lo\`{e}ve expansions of stochastic processes.

A “Luminous” Solution to the Basel Problem

Summary Years ago, D. Dumont gave a beautiful heuristic proof relating two famous equalities of Gregory–Leibniz and Euler. The purpose of this note is to make his proof rigorous.

A Wilf–Zeilberger–Based Solution to the Basel Problem With Applications

A Wilf–Zeilberger–Based Solution to the Basel Problem With Applications

Two short proofs of the formula

Euler’s striking proof of the fact that the infinite sum ${1 \over {{1^2}}}\, + \,{1 \over {{2^2}}}\, + \,{1 \over {{3^2}}} + \,...\, + \,{1 \over {{n^2}}} + \,...\, = {{{\pi ^2}} \over 6}$ continues to charm all who become acquainted with it as well as to motivate the search for new proofs. With the introduction of the zeta function $\zeta (s)\zeta = \sum\limits_{n = 1}^\infty {{1 \over {{n^s}}}} $ , the above sum is rewritten as $\zeta (2)$ and the problem of its evaluation in closed form is often named after Basel - the city of Euler’s birth. Many solutions to the Basel problem exist today, with several original ones appearing on the pages of The Mathematical Gazette. The purpose of this Note is to present two short proofs, the first of which is a simplification of one in the literature and the second is believed to be new, of the equivalent fact that $\sum\limits_{n = 0}^\infty {{1 \over {{{(2n + 1)}^2}}} = {{{\pi ^2}} \over 8}} .$

Euler and the Cubic Basel Problem

In 1734, Leonhard Euler summed the infinite series of reciprocals of the squares, thereby solving a challenge known as the “Basel problem.” He later extended his method to find closed-form sums for the reciprocals of 4th, 6th, and other even powers. But those techniques did not yield a value for the sum of the reciprocals of the cubes. Here, we show how Euler tried to evaluate this series by transforming it into the sum of a strange constant and an even stranger integral.

SAG Cluster: An unsupervised graph clustering based on collaborative similarity for community detection in complex networks

Many real-world social networks such as brain graph, protein structure, food web, transportation system, World Wide Web, online social networks exist in the form of a complex network. In such complex networks, pattern identification or community detection requires extra effort in which identifying community is a significant problem in various research areas. Most of the clustering methods on graphs predominantly emphasize on the topological structure without considering connectivity between vertices and not bearing in mind the vertex properties/attributes or similarity-based on indirectly connected vertices. A novel clustering algorithm SAG-Cluster with K-medoids framework presented for detecting communities using a collaborative similarity measure which considers attribute importance in case the pair of disconnected nodes. A novel path strategy using classic Basel problem for the indirectly connected node as well as balanced attribute similarity and distance function is proposed. On two real data sets, experimental results show the effectiveness of SAG-Cluster with the comparison of other relevant methods.

The Basel Problem: A Physicist’s Solution

Some time ago Wastlund reformulated the Basel problem in terms of a physical system using the proportionality of the apparent brightness of a star to the inverse square of its distance. Inspired by this approach, we give another physical interpretation which, in our opinion, is simpler, natural enough, and very Eulerian in its spirit.

Modified Equations and the Basel Problem

Discretizations of differential equations are often studied through their modified equation. This is a differential equation, usually obtained as a power series, with solutions that exactly interpolate the discretization. By comparing the St\"ormer-Verlet discretization of the harmonic oscillator with its modified equation, we obtain a relatively simple derivation of the expansion \[ \left( \arcsin \frac{h}{2} \right)^2 = \frac{1}{2} \sum_{k=1}^\infty \frac{(k-1)!^2}{(2k)!} h^{2k}, \] which can be used to show that $\zeta(2) = \frac{\pi^2}{6}$.

Basel Problem: A Solution Motivated by the Power of a Point

We give a new solution of the famous Basel problem, inspired by Euclidean geometry.

Basel Problem

Summary A rigorous elementary proof of the Basel problem [6, 1] ∑ n = 1 ∞ 1 n 2 = π 2 6 $$\sum\nolimits_{n = 1}^\infty {{1 \over {n^2 }} = {{\pi ^2 } \over 6}} $$ is formalized in the Mizar system [3]. This theorem is item #14 from the “Formalizing 100 Theorems” list maintained by Freek Wiedijk at http://www.cs.ru.nl/F.Wiedijk/100/.

Basel Problem – Preliminaries

Summary In the article we formalize in the Mizar system [4] preliminary facts needed to prove the Basel problem [7, 1]. Facts that are independent from the notion of structure are included here.

A Short and Elementary Proof of the Basel Problem

SummaryBy slightly changing a beautiful and little-known argument by E. L. Stark, we give a short and elementary proof of the celebrated Basel problem.

Special values of the Riemann zeta function via arcsine random variables

In this paper, using arcsine variables, we get a new elementary proof of ζ(2)=π2/6, known as the Basel problem, and the Euler formula. Using exponential variables, we get an Euler-like formula. We can also solve the Basel problem by using Wigner’s semicircle law and the Legendre generating function.

O PROBLEMA QUE TORNOU EULER FAMOSO

http://dx.doi.org/10.5902/2179460X12973In this study we present the Basel Problem whose answer has Leonhard Euler famous. Presents the Euler’s Proof which long contained a passage regarded as incorrect, but that after the Weierstrass Factorization Theorem was accepted. Also addresses other resolutions to the problem, its geometric representation and an application. It also discusses, through this problem, the importance of the History of Mathematics for the teacher of this discipline.