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

200 INTRODUCTION TO MATHEMATICS
proceed to explain what the "limit" of a
series is.
The summation of a series approximates to
a limit when the sum of any number of its
terms, provided the number be large enough,
is as nearly equal to the limit as you care to
approach. But this description of the meaning of approximating to a limit evidently will
not stand the vigorous scrutiny of modern
mathematics. What is meant by large
enough, and by nearly equal, and by care to
approach? All these vague phrases must be
explained in terms of the simple abstract
ideas which alone are admitted into pure
mathematics.
Let the successive terms of the series be
U, U2, U3, U4,..., Un, etc., so that u, is the
nth term of the series. Also let Sn be the
sum of the Ist n terms, whatever n may be.
So thatSi =U, s2=Ui+u2, s3s-u+u2+u3, and
Sn=U1+U2+U3+... +Un.
Then the terms si, 2S, s,... n,... form
a new series, and the formation of this series
is the process of summation of the original
series. Then the "approximation" of the
summation of the original series to a "limit"
means the "approximation of the terms of
this new series to a limit." And we have