Finding the sum of any series from a given general term
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+1 – Bn+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.