De summis serierum reciprocarum

English Title

On the sums of series of reciprocals

Authors

Leonhard Euler

Enestrom Number

41

Fuss Index

174

Original Language

Latin

Archive Notes

According to Eneström, this paper was not handed in after its presentation at the St. Petersburg Academy.

Content Summary

Euler finds the values of ζ(2n), where ζ is now named as the Riemann zeta function. He introduces the Euler numbers E_n. All the odd-indexed Euler numbers are zero, E₀=1, E₂=-1, E₄=5, E₆=-61, E₈=1380, and they satisfy (E+1)ⁿ + (E-1)ⁿ = 0. (The Euler numbers are the coefficients of the series expansion of sech(x) and are related to Bernoulli numbers.) Among other things, this paper includes an infinite product formula for sin(x)/x.

Published as

Journal article

Published Date

1740

Written Date

1735

Original Source Citation

Commentarii academiae scientiarum Petropolitanae, Volume 7, pp. 123-134.

Opera Omnia Citation

Series 1, Volume 14, pp.73-86.

Record Created

2018-09-25