Misc,
On the sums of series of reciprocals
(February 2008)
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.
