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On the Summation of Divergent Series

Published online by Cambridge University Press:  18 August 2016

A. De Morgan
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Extract

In the last Number I gave the most elementary view I could arrive at of Arbogast's method of development. In the communication following I saw that Mr. Peter Gray had referred to Stirling's theorem; and this suggested that it might be useful to give, by means of common algebra only, an account of the two most important cases of summation of many terms of a divergent series.

Type
Research Article
Information
Journal of the Institute of Actuaries , Volume 12 , Issue 5 , October 1865 , pp. 245 - 252
Copyright
Copyright © Institute and Faculty of Actuaries 1866

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References

page 246 note * This ingenious proof was given me, 37 years ago, by a pupil of the age of 13, whose mathematical power was singularly in advance of his years. Of many things as worthy of remark in one so young, I only remember what is here given. Time and thought have developed this boy into Professor Sylvester, whose inventive power, in everything to which his taste has led him, places him in the highest rank.

The divergence of the series was first noticed and proved by John Bernoulli, and another proof was given by James Bernoulli. Both are much, more difficult than that by collection into lots each greater than half a unit. I do not know who first gave this.

page 246 note † In this Journal, supported by contributors who have constantly to think of the common logarithm, it is very common to distinguish the Naperean logarithm when it is used. But it is now so well established, in algebraical writing, that log x shall mean the Naperean logarithm, that it would be a good thing if, without further mention, the common logarithm were always denoted by c. log x.

page 251 note * , Todhunter Hist. of Probability, p. 65.Google Scholar This work will be very useful to students who want to make a sound and complete preparation for the profession of an actuary.