##
E47 -- Inventio summae cuiusque seriei ex dato termino generali

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

**Summary:**

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 0^{n} + 1^{n} + 2^{n} +
3^{n} + ... +
(*x* - 1)^{n}, (*x*=1, 2, 3, ...) and gets the relationship on Bernoulli numbers that
(*B*+1)^{n+1} - *B*^{n+1} = 0.
He gets an infinite series approximation for the *n*th partial sum of the harmonic series.

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

**Publication:**
- Originally published in
*Commentarii academiae scientiarum Petropolitanae* 8, 1741, pp. 9-22
*Opera Omnia*: Series 1, Volume 14, pp. 108 - 123
- Reprinted in
*Comment. acad. sc. Petrop*. 8, ed. nova, Bononiae 1752, pp. 7-19 [47a]

**Documents Available:**
- Original publication: E047
(in the
*Commentarii*)
- English Translation (Jordan Bell): E47
- German Translation (Alexander Aycock and Arseny Skryagin): E047

- The Euler Archive attempts to monitor current scholarship for articles and books that may be of interest to Euler Scholars. Selected references we have found that discuss or cite E47 include:
- Calinger R., “Leonhard Euler: The first St Petersburg years (1727-1741).” Historia Mathematica, 23 (2), pp. 121-166 (May 1996).
- Ferraro G., “Functions, functional equations, and the laws of continuity in Euler.” Historia Mathematica, 27 (2), pp. 107-132 (May 2000).
- Ferraro G., “Some aspects of Euler's theory of series: Inexplicable functions and the Euler-Maclaurin summation formula.” Historia Mathematica, 25 (3), pp. 290-317 (Aug 1998).
- Grabiner JV., “Was Newton's calculus a dead end? The continental influence of Maclaurin's treatise of fluxions.” American Mathematical Monthly, 104 (5), pp. 393-410 (May 1997).

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