Finding the sum of any series from a given general term

TL;DR

This paper explores Euler's derivation of the Euler-Maclaurin summation formula and demonstrates its application to summing powers, harmonic series, and approximating zeta function values, highlighting its significance in analysis.

Contribution

It provides a detailed analysis of Euler's original derivation of the summation formula and applies it to various series, including powers and zeta functions, with explicit calculations.

Findings

Derived the Euler-Maclaurin summation formula from Taylor expansion.

Explicit formulas for sums of powers up to n=16.

Approximations for harmonic series and zeta function values.

Abstract

Translation from the Latin original, "Inventio summae cuiusque seriei ex dato termino generali" (1735). E47 in the Enestrom index. In this paper Euler derives the Euler-Maclaurin summation formula, by expressing y(x-1) with the Taylor expansion of y about x. In sections 21 to 23 Euler uses the formula to find expressions for the sums of the nth powers of the first x integers. He gives the general formula for this, and works it out explicitly up to n=16. In sections 25 to 28 he applies the summation formula to getting approximations to partial sums of the harmonic series, and in sections 29 to 30 to partial sums of the reciprocals of the odd positive integers. In sections 31 to 32, Euler gets an approximation to zeta(2); in section 33, approximations for zeta(3) and zeta(4). I found David Pengelley's paper "Dances between continuous and discrete: Euler's summation formula", in the MAA's "Euler at 300: An Appreciation", edited by Robert E. Bradley, Lawrence A. D'Antonio, and C. Edward Sandifer, very helpful and I recommend it if you want to understand the summation formula better.