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

SERIES 201
now to explain what we mean by the approximation to a limit of the terms of a series.
Now, remembering the definition (given in
Chapter XII) of a standard of approximation, the idea of a limit means this: 1 is
the limit of the terms of the series si, s2,
s3,... Sn,..., if, corresponding to each
real mumber k, taken as a standard of
approximation, a term s, of the series can
be found so that all succeeding terms (i.e.
Sn+l, Sn+2, etc.) approximate to 1 within
that standard of approximation. If another
smaller standard k1 be chosen, the term
s, may be too early in the series, and a
later term Sm with the above property will
then be found.
If this property holds, it is evident that as
you go along to series si, S2, 83,...,,Sn...
from left to right, after a time you come to
terms all of which are nearer to I than any
number which you may like to assign. In
other words you approximate to I as closely
as you like. The close connection of this
definition of the limit of a series with the
definition of a continuous function given in
Chapter XI will be immediately perceived.
Then coming back to the original series ul,
U2, Us,..., Un..., the limit of the terms of
the series si, 2, Sn...,,..., is called
the "sum to infinity" of the original series.
But it is evident that this use of the word