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

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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).

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