An elementary view of Euler’s summation formula. (English) Zbl 1076.41509

Summary: Euler’s summation formula, which relates integrals and finite sums, is widely used in numerical analysis, analytic number theory, and the theory of asymptotic expansions. It contains Bernoulli numbers and periodic Bernoulli functions and is ordinarily discussed in courses in advanced calculus or real and complex analysis. Starting with a simple diagram, this paper shows how the general formula can be discovered using only elementary calculus, and it also shows how Bernoulli numbers and Bernoulli functions arise naturally along the way. The formula is used to calculate the first 8 digits in Euler’s constant.


41A55 Approximate quadratures
40A25 Approximation to limiting values (summation of series, etc.)
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