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

SERIES 197
number of terms (say n terms) has the same
characteristics as far as next-door neighbours
are concerned as an infinite series; it only
differs from infinite series in having a last
ierm, namely, the nth.
The important thing to do with a series of
numbers-using for the future "series" in
the restricted sense which has just been
mentioned-is to add its successive terms
together.
Thus if ui, 2, u3,... Un.. are respectively the Ist, 2nd, 3rd, 4th,... nth,...
terms of a series of numbers, we form successively the series u1, Ui+u2, u1+u2+u3, Ui+
U2+U3+U4, and so on; thus the sum of the
Ist n terms may be written.
Ul+U2+u3+... +Un.
If the series has only a finite number of
terms, we come at last in this way to the
sum of the whole series of terms. But, if
the series has an infinite number of terms,
this process of successively forming the sums
of the terms never terminates; and in this
sense there is no such thing as the sum of an
infinite series.
But why is it important successively to add
the terms of a series in this way? The answer
is that we are here symbolizing the fundamental mental process of approximation.
This is a process which has significance far