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Divergence rules OK?

Published online by Cambridge University Press:  22 September 2016

J. P. Coleman
J. P. Coleman*
Affiliation:
Department of Mathematics, University of Durham, DH1 3LE
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Extract

The Norwegian mathematician Abel wrote in 1826 [1] “On the whole, divergent series are a devilry, and it is a shame to base any demonstration upon them”. At the other extreme Heaviside, a British physicist and engineer whose unorthodox methods raised many mathematical eyebrows, is quoted [2] as having said “The series is divergent; therefore we may be able to do something with it”. Heaviside may be accused of exaggeration but Abel, though his attitude was very reasonable in the absence of a proper theory of divergent series, was somewhat over-zealous in his total condemnation of the use of such series. Divergent series can be useful and in some cases are of far greater practical value than their convergent counterparts.

Type
Research Article
Information
The Mathematical Gazette , Volume 64 , Issue 429 , October 1980 , pp. 180 - 184
Copyright
Copyright © Mathematical Association 1980

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References

1. Ore, O., Niels Henrik Abel. University of Minnesota Press (1957). Much the same statement appears in French in Bjerknes, C. A., Niels-Henrik Abel (Gauthier-Villars, Paris 1885) and in Ch. Pesloùan, L. de, N.-H. Abel, Sa vie et son oeuvre (Gauthier-Villars, Paris 1906), whereas a slightly different version is to be found in Oeuvres completes de Niels Henrik Abel, Nouvelle edition par mm. L. Sylowet S. Lie (Oslo 1881).Google Scholar
2. Kline, M., Mathematical thought from ancient to modern times. Oxford University Press (1972).Google Scholar
3. Olver, F. W. J., Asymptotics and special functions. Academic Press (1974).Google Scholar