Page:Philosophical Transactions - Volume 145.djvu/192

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173
mr. w.h.l. russell on the theory of definite integrals.

same factorials, so that we can deduce the value of many definite integrals from one series.

I shall now give an example of the summation of a factorial series of a somewhat different nature.

Consider the series—

we know that

Hence by substitution the above series becomes

There are other series of an analogous nature which may be summed in a similar manner: the object of introducing the above summation in this paper, is to point out the use of the integral , when impossible factors occur in the denominators of the successive terms of a factorial series.

In the 'Exercices de Mathématiques,' Cauchy has proved that if be a quantity of the form , and continually approach zero as indefinitely increases whatever be , then the residue of is equal to zero, the limits of being 0 and (), and those of , and . From this theorem he deduces the sums of certain series, which I shall presently consider; but must first give certain results which will be useful in the sequel.

Since

Again, since
we find
whence we have