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Summation of slowly convergent series

Published online by Cambridge University Press:  24 October 2008

T. M. Cherry
T. M. Cherry
Affiliation:
The UniversityMelbourneAustralia
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Extract

Let ∑un be a convergent infinite series which is not summable in finite form. In principle its sum can be found, to within any preassigned error ε, by adding numerically a sufficient number of terms; but if the series is slowly convergent, the ‘sufficient number’ of terms may be prohibitively large. A plan to deal with this case is to separate the series into a ‘main part’ u0+u1+ … +un−1 and a ‘remainder’ Rn = un+un+1+…; the main part is evaluated by direct summation, while the remainder is transformed analytically into a series which is more rapidly ‘convergent’, in the practical sense, and so evaluated. For example, the Euler-Maclaurin sum-formula gives such a transformation. It commonly happens that the new form of the remainder Rn is a divergent series, but that it represents Rn asymptotically as n ˜ ∞. It is for this reason that the transformation is applied to Rn instead of to the whole series; for practical use we have to choose n sufficiently large for the error inherent in the use of the asymptotic series to be below the preassigned bound ε.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 46 , Issue 3 , July 1950 , pp. 436 - 449
Copyright
Copyright © Cambridge Philosophical Society 1950

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References

* I have searched the Enzyklopädie; the treatises on Infinite Series of Bromwich, Knopp and Fort; those on Finite Differences of Nörlund, Milne-Thompson and Fort; and those on Numerical Methods of Whittaker and Robinson, Runge and Willers. The formula (7), in the formal case p = ∞, is in Boole, , Calculus of finite differences, 2nd ed. (1872), p. 85Google Scholar, Ex. 12, with a reference via Gudennann to Euler.

* Here, and below, we simplify the notation by writing D in place of D t and D n. There will be no confusion, since øn is a function of t only and f(n) of n only.

† See, for example, Bromwich, Infinite aeries (2nd ed.), §24.

* The Laplace-integral representation is required only for the discussion of the error-terms in §§ 3, 4.

* About twenty tests of this rough estimate have been made on the non-trivial example of §6. In no case was the estimate deficient by more than a factor of 1·2; and usually it was not deficient at all.

* The function B s, n(t) rather than t nB s, n(t) is taken as fundamental because it submits more readily to tabular interpolation.

* This value of t was chosen with a view to the example of § 6, case 3.