#### Title

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

#### Enestrom Number

61

#### Fuss Index

175

#### Original Language

Latin

#### Published as

Journal article

#### Published Date

1743

#### Written Date

1742

#### Content Summary

Euler gives an infinite product for sin(*x*)/*x* as well as the formulas sin(*x*) = (*e ^{ix}* –

*e*)/(2

^{-ix}*i*) and cos(

*x*) = (

*e*+

^{ix}*e*)/2, which he had communicated with Goldbach on 9 December 1741 and 8 May 1742. He presents an infinite series for Ï /sin(Ï

^{-ix}*s*) and for Ï ∙cot(Ï

*s*), an evaluation of the sums of the reciprocals of odd squares as Ï

^{2}/8, and an evaluation of the corresponding alternating sum as Ï

^{2}/(8â2).

#### 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