On the sums of series of reciprocals

TL;DR

Euler's classical work provides an exact expression for the sum of reciprocals squared, revealing that it equals pi squared over six, and extends to other even zeta values using polynomial root-coefficient relationships.

Contribution

This paper presents Euler's original derivation of the sum of reciprocals of squares and related zeta values, connecting polynomial identities with trigonometric series.

Findings

Sum of reciprocals squared equals pi^2/6

Euler extends the method to other even zeta values

Provides explicit formulas involving Bernoulli numbers

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.