De summa seriei ex numeris primis formatae 1/3 – 1/5 + 1/7 + 1/11 – 1/13 – 1/17 + 1/19 + 1/23 – 1/29 + 1/31 etc. ubi numeri primi formae 4n–1 habent signum positivum, formae autem 4n+1 signum negativum

English Title

On the sum of the series of numbers of the form 1/3 – 1/5 + 1/7 + 1/11 – 1/13 – 1/17 + 1/19 + 1/23 – 1/29 + 1/31 etc. in which the prime numbers of the form 4n–1 have positive signs, and those of the form 4n+1 have negative signs

Authors

Leonhard Euler

Enestrom Number

596

Fuss Index

186

Original Language

Latin

Content Summary

First, Euler notes that the sum of the reciprocals of the primes diverges, as does the "logarithmic sum" or harmonic series. Then, his derivations begin with the Leibniz series, 1 − 1/3 + 1/5 − 1/7 + 1/9 ± ⋯ = π/4.

Published as

Collection article

Published Date

1785

Written Date

1775

Original Source Citation

Opuscula analytica, Volume 2, pp. 240-256.

Opera Omnia Citation

Series 1, Volume 4, pp.146-162.

Record Created

2018-09-25