Basel problem

Abstract: The celebrated Basel Problem, that of finding the infinite sum 1 + 1/4 + 1/9 + 1/16 + …, was open for 91 years. In 1735 Euler showed that the sum is π2/6. Dozens of other solutions have been found....

Abstract: Probabilistic proofs and interpretations are given for the q-binomial theorem, q-binomial series, two of Euler's fundamental partition identities, and for q-analogs of product expansions for the Riemann zeta and Euler phi functions. The underlying processes involve Bernoulli trials with variable probabilities. Also presented are several variations on the classical derangement problem inherent in the distributions considered.

Abstract: Introduction Leonhard Euler, the decade 1750-1760 Rudiger Thiele Euler's fourteen problems C. Edward Sandifer The Euler archive: giving Euler to the world Dominic Klyve and Lee Stemkoski The Euler-Bernoulli proof of the fundamental theorem of algebra Christopher Baltus The quadrature of Lunes, from Hippocrates to Euler Stacy G. Langton What is a function? Rudiger Thiele Enter, stage center: the early drama of the hyperbolic functions Janet Heine Barnett Euler's solution of the Basel problem - the longer story C. Edward Sandifer Euler and elliptic integrals Lawrence D'Antonio Euler's observations on harmonic progressions Mark McKinzie Origins of a classic formalist argument: power series expansions of the logarithmic and exponential functions Mark McKinzie Taylor and Euler: linking the discrete and continuous Dick Jardine Dances between continuous and discrete: Euler's summation formula David J. Pengelley Some combinatorics in Jacob Bernoulli's Ars Conjectandi Stacy G. Langton The Genoese lottery and the partition function Robert E. Bradley Parallels in the work of Leonhard Euler and Thomas Clausen Carolyn Lathrop and Lee Stemkoski Three bodies? Why not four? The motion of the Lunar Apsides Robert E. Bradley 'The fabric of the universe is most perfect': Euler's research on elastic curves Lawrence D'Antonio The Euler advection equation Roger Godard Euler rows the boa C. Edward Sandifer Lambert, Euler, and Lagrange as map makers George W. Heine, III Index.

Abstract: This translation has been published in Stephen Hawking (ed.), "God Created the Integers", published in 2007 by Running Press. There may have been some changes to the final published version and this copy. This is a translation from the Latin original, "De summis serierum reciprocarum" (1735). E41 in the Enestrom index. In this paper Euler finds an exact expression for the sum of the squares of the reciprocals of the positive integers, namely pi^2/6. He shows this by applying Newton's identities relating the roots and coefficients of polynomials to the power series of the sine function. Indeed, in other words this result is zeta(2)=pi^2/6, and Euler also works out zeta(4),zeta(6),...,zeta(12). His method will work out zeta(2n) for all n, but he does not give a general expression for zeta(2n); he gives a general expression involving the Bernoulli numbers in a latter paper.