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Most Power Series have Radius of Convergence 0 or 1*

Published online by Cambridge University Press:  20 November 2018

J. J. F. Fournier  and
P. M. Gauthier
J. J. F. Fournier
Affiliation:
University of British Columbia
P. M. Gauthier
Affiliation:
Université de Montréal
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Consider a random power series Σ0∞ cn zn, that is, with coefficients {cn}0∞ chosen independently at random from the complex plane. What is the radius of convergence of such a series likely to be?

One approach to this question is to let the {cn}0∞ be independent random variables on some probability space. It turns out that, with probability one, the radius of convergence is constant. Moreover, if the cn are symmetric and have the same distribution, then the circle of convergence is almost surely a natural boundary for the analytic function given by the power series (See [1, Ch. IV, Section 3]). Our treatment of the question will be elementary and will not use these facts.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 18 , Issue 1 , April 1975 , pp. 39 - 40
Copyright
Copyright © Canadian Mathematical Society 1975

Footnotes

*

Research supported by N.R.C. grants A-4822 and A-5597, and by a grant from the Quebec Government.

References

1. Kahane, J.-P., Some random series of functions, Heath, 1968.Google Scholar