E63 -- Demonstration de la somme de cette suite 1 + 1/4 + 1/9 + 1/16 + etc.

E63 -- Demonstration de la somme de cette suite \( 1 + \frac{1}{4} + \frac{1}{9} + \frac{1}{16} +\) etc.

(Demonstration of the sum of the following series: \( 1 + \frac{1}{4} + \frac{1}{9} + \frac{1}{16} +\) etc.)

Summary:

The answer hasn't changed; it's still \(\frac{\pi^2}{6}\). He does the sum of the odd square reciprocals by an arcsine integral, and considers the differential equation \((1-x^2)y''-xy'-1=0\).

Publication:

Originally published in Journ. lit. d'Allemange, de Suisse et du Nord, 2:1, 1743, p. 115-127.
Opera Omnia: Series 1, Volume 14, pp. 177 - 186
Reprinted in Bibl. math. 83, 1907/8, pp. 54-60 [E63a]

Documents Available:

Original Document: E63
The Euler Archive attempts to monitor current scholarship for articles and books that may be of interest to Euler Scholars. Selected references we have found that discuss or cite E63 include:
- Sandifer E., "The Basel Problem with integrals." How Euler Did It. (Published online by the MAA.) Sandifer gives a thorough treatment of Euler's solution to the Basel Problem from both E63 and E41.

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