E47 -- Inventio summae cuiusque seriei ex dato termino generali

(Finding the sum of any series from a given general term)


Euler continues with the methods of E25 to attack z(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 on Bernoulli numbers that (B+1)n+1 - Bn+1 = 0. He gets an infinite series approximation for the nth partial sum of the harmonic series.

According to the records, it was presented to the St. Petersburg Academy on October 13, 1735.

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