Skip to main content
Library homepage
 
Loading table of contents menu...
Physics LibreTexts

10.6: F- The Euler-MacLaurin Formula and Asymptotic Series

  • Page ID
    6397
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

    You know that a sum can be approximated by an integral. How accurate is that approximation? The Euler-MacLaurin formula gives the corrections.

    \(\sum_{k=0}^{n-1} f(k) \approx \int_{0}^{n} f(x) d x\)

    \( -\frac{1}{2}[f(n)-f(0)]+\frac{1}{12}\left[f^{\prime}(n)-f^{\prime}(0)\right]-\frac{1}{720}\left[f^{\prime \prime \prime}(n)-f^{\prime \prime \prime}(0)\right]+\frac{1}{30240}\left[f^{(v)}(n)-f^{(v)}(0)\right]-\frac{1}{1209600}\left[f^{(\mathrm{vii})}(n)\right.\)

    This series is asymptotic. If the series is truncated at any point, it can give a highly accurate approximation. But the series may be either convergent or divergent, so adding additional terms to the truncated series might give rise to a poorer approximation. The Stirling approximation is a truncation of an asymptotic series.

    References

    C.M. Bender and S.A. Orszag, Advanced Mathematical Methods for Scientists and Engineers, (McGraw-Hill, New York, 1978).

    M. Abramowitz and I. Stegum, Handbook of Mathematical Functions, (National Bureau of Standards, Washington, D.C., 1964).


    This page titled 10.6: F- The Euler-MacLaurin Formula and Asymptotic Series is shared under a CC BY-SA license and was authored, remixed, and/or curated by Daniel F. Styer.

    • Was this article helpful?