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Note on a Divergent Series

Published online by Cambridge University Press:  24 October 2008

G. H. Hardy
G. H. Hardy
Affiliation:
Trinity CollegeCambridge
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Extract

1. The series

is the simplest and most familiar power series whose radius of convergence is zero. It is natural to regard it as a development of the function G(z) defined, when

by

For

say;and

or

according as x is positive or negative. Thus the series (1·1) is an asymptotic series for G(z) in the sense of Poincaré.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 37 , Issue 1 , January 1941 , pp. 1 - 8
Copyright
Copyright © Cambridge Philosophical Society 1941

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References

* | 1 − zt|2 = 1 − 2rt cos θ + r 2t 2 has the minimum sin2θ or 1, according as cos θ is positive or negative.

* The point is to choose λn so that

is an integral function, for every positive δ, but with “little to spare”.

* That is to say, of the distant part of C′. The inequality is not satisfied near u = A.

* P(∞, η) is the region which is the limit of P(R, η) when R → ∞.

* We express a n first by an integral round a contour lying inside the circle of convergence of the series (7·1). We may deform this contour into C because f(z) is regular in P and satisfies (7·2).

† This is the definition of g k+1(z).