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

202 INTRODUCTION TO MATHEMATICS
"sum" is very aritficial, and we must not
assume the analogous properties to those of
the ordinary sum of a finite number of terms
without some special investigation.
Let us look at an example of a "sum to
infinity." Consider the recurring decimal
1111.... This decimal is merely a way of
symbolizing the "sum to infinity" of the
series.1,.01,.001, -0001, etc. The corresponding series found by summation is si=-l,
s2=l11, s3=111*, s4=1111, etc. The limit
of the terms of this series is '; this is easy to
see by simple division, for
-1 +-9 =.11 + *11+0lO0 1= etc.
Hence, if - is given (the k of the definition),
*1 and all succeeding terms differ from - by
less than -1; if -ol is given (another choice
for the k of the definition),.111 and all succeeding terms differ from 1 by less than — o;
and so on, whatever choice for k be made.
It is evident that nothing that has been
said gives the slightest idea as to how the
"sum to infinity" of a series is to be
found. We have merely stated the conditions which such a number is to satisfy. Indeed, a general method for finding in all
cases the sum to infinity of a series is intrinsically out of the question, for the simple reason
that such a "sum," as here defined, does not
always exist. Series which possess a sum to