E521 -- Extraits de différentes lettres de M. Euler à M. le Marquis de Condorcet
(Extracts from different letters of Mr. Euler to Mr. le Marquis de Condorcet)
This paper consists of extracts of letters written by Euler to Condorcet. The letters are about a couple of integrals and a series related to the binomial coefficients. Euler evaluates the integrals by finding a function \(Q(x,y)\) such that integrating \(Q\) first with respect to \(x\), then \(y\), gives the integral he wants to evaluate. He then reverses the order of integration in order to evaluate the integral. Then he shows how he originally obtained the result through clever series manipulation. Finally he demonstrates a result about the sum of the squares of the binomial coefficients and uses this to obtain a series converging to \(4/\pi\) by allowing fractions to interpolate his binomial result.
Originally published in Memoires de l'academie des sciences de Paris 1778, 1781, pp. 603-614
Opera Omnia: Series 1, Volume 18, pp. 69 - 82
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