An introduction to mathematics, by A. N. Whitehead.

SERIES 205
perties of the sum of a finite number of terms
to that of the sum of an infinite series. For
the most elementary property of a finite
number of terms is that of course they
possess a sum: but even this fundamental
property is not necessarily possessed by an
infinite series. This caution merely states
that we must not be misled by the suggestion
of the technical term "sum of an infinite
series." It is usual to indicate the sum of
the infinite series
Ui, U, U3,... Un... by
Ul +U2 +U3+...+ +...
We now pass on to a generalization of the
idea of a series, which mathematics, true to
its method, makes by use of the variable.
Hitherto, we have only contemplated series
in which each definite term was a definite
number. But equally well we can generalize,
and make each term to be some mathematical
expression containing a variable x. Thus
we may consider the series 1,,, x, 3...,
x",..., and the series
X2 X3 Xn
x, 2' *,n *.
In order to symbolize the general idea of
any such function, conceive of a function of
z, f,(x) say, which involves in its formation
a variable integer n, then, by giving n the