#### English Title

On the sums of series of reciprocals

#### Enestrom Number

41

#### Fuss Index

174

#### Original Language

Latin

#### English Summaries

Euler finds the values of ζ(2*n*), 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,

*E*

_{0}=1,

*E*

_{2}=-1,

*E*

_{4}=5,

*E*

_{6}=-61,

*E*

_{8}=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

#### 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 Î¶(2*n*), where Î¶ is now named as the Riemann zeta function. He introduces the Euler numbers *E _{n}*. All the odd-indexed Euler numbers are zero,

*E*

_{0}=1,

*E*

_{2}=-1,

*E*

_{4}=5,

*E*

_{6}=-61,

*E*

_{8}=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*.

#### 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

## Notes

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