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.