De summis serierum reciprocarum ex potestatibus numerorum naturalium ortarum dissertatio altera, in qua eaedem summationes ex fonte maxime diverso derivantur

English Title

On sums of series of reciprocals from powers of natural numbers from another discussion, in which the sums are derived principally from another source

Authors

Leonhard Euler

Enestrom Number

61

Fuss Index

175

Original Language

Latin

Content Summary

Euler gives an infinite product for sin(x)/x as well as the formulas sin(x) = (e^ix – e^-ix)/(2i) and cos(x) = (e^ix + e^-ix)/2, which he had communicated with Goldbach on 9 December 1741 and 8 May 1742. He presents an infinite series for π/sin(πs) and for π∙cot(πs), an evaluation of the sums of the reciprocals of odd squares as π²/8, and an evaluation of the corresponding alternating sum as π²/(8√2).

Published as

Journal article

Published Date

1743

Written Date

1742

Original Source Citation

Miscellanea Berolinensia, Volume 7, pp. 172-192.

Opera Omnia Citation

Series 1, Volume 14, pp. 138-155.

Record Created

2018-09-25