On the sums of series of reciprocals
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.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 June 2005
 arXiv:
 arXiv:math/0506415
 Bibcode:
 2005math......6415E
 Keywords:

 Mathematics  History and Overview;
 Mathematics  Number Theory;
 01A50;
 11M06;
 1103;
 11B68
 EPrint:
 8 pages, 1 figure