Extracts of different letters of Mr. Euler to Mr. le Marquis de Condorcet
This paper consists of extracts of letters written by Euler to Condorcet, which concern a couple of integrals and a series related to the binomial coefficients. First, Euler evaluates the integrals by finding a function Q(x,y) so that integrating Q first with respect to x, then with respect to y, gives the integral he wants to evaluate. He then reverses the order of integration in order to evaluate the integral. Next, 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/π by allowing fractions to interpolate his binomial result.
Original Source Citation
Mémoires de l'académie royale des sciences de Paris, Volume 1778, pp. 603-614.
Opera Omnia Citation
Series 1, Volume 18, pp.69-82.