English Title

Finding the sum of any series from a given general term

Authors

Leonhard Euler

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 0n + 1n + 2n + 3n + ... + (x-1)n, (x = 1, 2, 3, ...) and gets the relationship (B+1)n+1Bn+1 = 0 for Bernoulli numbers. He gets an infinite series approximation for the nth 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 ζ(2) for a second time. He starts with a Taylor series, builds a "Bernoulli polynomial" and uses it to evaluate 0n + 1n + 2n + 3n + ... + (x-1)n, (x = 1, 2, 3, ...) and gets the relationship (B+1)n+1Bn+1 = 0 for Bernoulli numbers. He gets an infinite series approximation for the nth 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

Included in

Mathematics Commons

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