从给定通项求任意级数的和
Finding the sum of any series from a given general term
AI总结：
欧拉通过泰勒展开推导欧拉-麦克劳林求和公式，并应用于整数幂和、调和级数部分和及zeta函数近似。
摘要：
翻译自拉丁文原版“Inventio summae cuiusque seriei ex dato termino generali”（1735年）。Enestrom索引编号E47。本文中，欧拉通过将y(x-1)用y关于x的泰勒展开表示，推导出欧拉-麦克劳林求和公式。在第21至23节，欧拉利用该公式求前x个整数的n次幂之和的表达式。他给出了这一般公式，并具体计算到n=16。在第25至28节，他将求和公式应用于调和级数部分和的近似，在第29至30节应用于正奇数倒数的部分和。在第31至32节，欧拉得到zeta(2)的近似值；在第33节，得到zeta(3)和zeta(4)的近似值。我发现David Pengelley的论文“Dances between continuous and discrete: Euler's summation formula”（收录于MAA的“Euler at 300: An Appreciation”，由Robert E. Bradley、Lawrence A. D'Antonio和C. Edward Sandifer编辑）非常有帮助，如果你想更好地理解求和公式，我推荐阅读。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
