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Sir, Having lately discovered some very easy methods of investigating the sums of certain infinite series, I have taken the liberty of requesting the favour of you to present them to the Royal Society. I have divided the subject into three parts: the first contains a new and general method of finding the sum of those series which De Moivre has found in one or two particular cases; but whose method, although it be in appearance general, will, upon trial, be found to be absolutely impracticable. The second contains the summation of certain series, the last differences of whose numerators become equal to nothing. The third contains observations on acorrection which is necessary in investigating the sums of certain series by collecting two terms into one, with its application to a variety of cases.

XXV. A new method of investigating the sums of infinite series. By the Rev. S. Vince, A.M. of Cambridge, in a letter to Henry Maty, A.M. Secretary