#### English Title

Finding the sum of any series from a given general term

#### Enestrom Number

47

#### Fuss Index

172

#### Original Language

Latin

#### English Summaries

Euler continues with the methods of E25 to attack ζ(2) for a second time. He starts with a Taylor series, builds a "Bernoulli polynomial" and uses it to evaluate 0^{n} + 1^{n} + 2^{n} + 3^{n} + ... + (*x*-1)^{n}, (*x =* 1, 2, 3, ...) and gets the relationship (*B*+1)^{n+1} – *B*^{n+1} = 0 for Bernoulli numbers. He gets an infinite series approximation for the *n*th partial sum of the harmonic series.

#### Published as

Journal article

#### Published Date

1741

#### Written Date

1735

#### Content Summary

Euler continues with the methods of E25 to attack Î¶(2) for a second time. He starts with a Taylor series, builds a "Bernoulli polynomial" and uses it to evaluate 0^{n} + 1^{n} + 2^{n} + 3^{n} + ... + (*x*-1)^{n}, (*x =* 1, 2, 3, ...) and gets the relationship (*B*+1)^{n+1} – *B*^{n+1} = 0 for Bernoulli numbers. He gets an infinite series approximation for the *n*th partial sum of the harmonic series.

#### Original Source Citation

Commentarii academiae scientiarum Petropolitanae, Volume 8, pp. 9-22.

#### Opera Omnia Citation

Series 1, Volume 14, pp.108-123.

#### Record Created

2018-09-25