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Formulæ for Powers and Reversion of Series

Published online by Cambridge University Press:  03 November 2016

Albert Eagle
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Extract

The formulæ for the powers and the reversion of any given series are not so readily accessible to those who are dependent upon modern mathematical text-books as they should be considering the frequent and manifold uses to which such formulæ can be put. I thought therefore that the following compendium of such formulae would be useful to readers of the Mathematical Gazette. The chief formulae in this paper are given in De Morgan’s Differential and Integral Calculus, but this book must be getting increasingly unavailable to ordinary readers except in reference libraries.

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Other
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The Mathematical Gazette , Volume 15 , Issue 211 , January 1931, pp. 285 - 291
Copyright
Copyright © Mathematical Association 1931

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* Here the language is being used which allows a maximum to be infinite. And unless both the neighbouring maxima are infinite, or in crude phrase unless there are no maxima at all, the reversed series can not be convergent for all values of y. That this obvious necessary condition is not sufficient is shown by the case of y = x 2 + x 4, for which the reversed series are not convergent for y > ¼. But in the case when y > ¼ we can get a convergent series for x in ascending powers of y -½ (all multiplied by y ¼) by writing the equation in the form y 4(l+x-2) and taking p = -¼ and q = ½ For any parabola of even, finite degree a series for x, convergent for large values of y, can always be obtained in this manner; the two series may, however, leave a large intermediate region of y in which neither of them is convergent.