# The Euler-MacLauren Formula as an Asymptotic Form of Poisson’s

• Henry E. Fettis
Chapter
Part of the ISNM 57: International Series of Numerical Mathematics / Internationale Schriftenreihe zur Numerischen Mathematik / Série internationale d’Analyse numérique book series (ISNM, volume 57)

## Abstract

The Euler-MacLauren formula
$$\int\limits_0^a {f(x)dx} = {I_t}(h) - \sum\limits_{i = 1}^m {{A_i}{h^{2i}} + {R_m}{h^{2m}}}$$
(1a)
where
$${A_i} = \frac{{{B_{2i}}}}{{2i!}}\left[ {{f^{(2i - 1)}}(nh) - {f^{(2i - 1)}}(0)} \right];\quad h = \frac{a}{n};$$
(1b)
Bn are the Bernoulli numbers, and It(h) is the trapezoidal sum:
$${R_m} = \frac{1}{{2m!}}{B_{2m}}{f^{(2m)}}(\xi ),\quad 0 \le \xi \le nh;$$
(1c)
can be found in nearly every text on numerical analysis (e.g., [1]-[5]). Its importance lies in the fact that it is the basis for the derivation of quadrature formulae of Newton-Cotes type, i.e., quadrature formulae which utilize values of the integrand at equally spaced values of “x”. As is well known, this formula is not convergent, but asymptotic, in the sense that, whereas
$${h^{2m}}{R_m}(h) \to 0$$
for h → O (m fixed), it does not hold that
$${h^{2m}}{R_m}(h) \to 0$$
for m → ∞ (h fixed). Its conventional derivation is long and involved, and the reader is frequently at a loss to know how it could have originally been arrived at.

## References

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