Abstract
The object of this paper is to explain two methods of finding the sums of a variety of infinite series. One of these the author discovered several years ago; but finding that some of the results to which it led were erroneous, he then declined publishing it. In inquiring into the causes of these errors, he was led to the second method, which employs the process of integration relative to finite differences. The cause of the fallacies in the former method was afterwards discovered, and in this paper a criterion is proposed for judging of the truth of the results, and a mode of correcting them where found to be erroneous. The sums of a variety of series are found by these methods; and the author concludes by observing, that he has since been informed by M. Poisson, that that gentleman had arrived at some nearly similar results in investigating a problem in physical astronomy, and also that some investigations of a similar nature were found amongst the papers of Lagrange, but that neither of these mathematicians had explained the cause of the errors, or given a method of correcting them.
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