seems almost a truism.
But, if we deduce E^n = (1 + [Delta])^n, [Delta]^n = (E-1)^n, and
expand by the binomial theorem and then operate on u0, we get the
general formulae

n.n - 1
un = u0 + n[Delta]u0 + ------- [Delta]^2u0 + ... + [Delta]^nu0,
1.2
n.n - 1
[Delta]^nu0 = u_n - nu_(n-1) + ------- u_(n-2) + ... + (-1)^nu0,
1.2

which are identical with the formulae in (ii.) and (i.) of S 3.

(iii.) What has been said under (ii.) applies, with certain
reservations, to the operations [Sigma] and [sigma], and to the
operation which represents integration. The latter is sometimes
denoted by D^-1; and, since [Delta][Sigma]un = un, and
[delta][sigma]u_n = u_n, we might similarly replace [Sigma] and
[sigma] by [Delta]^-1 and [delta]^-1. These symbols can be combined
with [Delta], E, &c. according to the ordinary laws of algebra,
provided that proper account is taken of the arbitrary constants
introduced by the operations D^-1, [Delta]^-1, [delta]^-1.

_Applications to Algebraical Series._

13. _Summation of Series._--If ur, denotes the (r+1)th term of a
series, and if vr is a function of r such that [Delta]v_r = u_r for
all integral values of r, then the sum of the terms u_m, u_(m+1), ...
un is v_(n+1) -v_m. Thus the sum of a number of terms of a series may
often be found by inspection, in the same kind of way that an integral
is found.

14. _Rational Integral Functions._--(i.) If u_r is a rational integral
function of r of degree p, then [Delta]ur, is a rational integral
function of r of degree p-1.

(ii.) A particular case is that of a _factorial_, i.e. a product of
the form (r+a+1) (r+a+2) ... (r+b), each factor exceeding the
preceding factor by 1. We have

[Delta].(r+a+1) (r+a+2) ... (r+b) = (b-a).(r+a+2) ... (r+b),

whence, changing a into a-1,

[Sigma](r+a+1)(r+a+2) ... (r+b) = _const._ + (r+a)(r+a+1) ...
(r+b)/(b-a+1).

A similar method can be applied to the series whose (r+1)th term is of
the form 1/(r+a+1) (r+a+2) ... (r+b).

(iii.) Any rational integral function can be converted into the sum of
a number of factorials; and thus the sum of a series of which such a
function is the general term can be found. For example, it may be
shown in this way that the sum of the pth powers of the first