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

SERIES 203
infinity are called convergent, and those which
do not possess a sum to infinity are called
divergent.
An obvious example of a divergent series
is 1, 2, 3,..., n... i.e. the series of integers in their order of magnitude. For
whatever number I you try to take as its
sum to infinity, and whatever standard of
approximation k you choose, by taking
enough terms of the series you can always
make their sum differ from I by more than
k. Again, another example of a divergent
series is 1, 1, 1, etc., i.e. the series of
which each term is equal to 1. Then the
sum of n terms is n, and this sum grows
without limit as n increases. Again, another
example of a divergent series is 1, -1, 1, -1,
1, -1, etc., i.e. the series in which the terms
are alternately 1 and -1. The sum of an
odd number of terms is 1, and of an even
number of terms is O. Hence the terms of
the series Si, 82, S3,... Sn,... do not approximate to a limit, although they do not
increase without limit.
It is tempting to suppose that the condition for u1, u2,..'. Un,... to have a sum
to infinity is that u, should decrease indefinitely as n increases. Mathematics would
be a much easier science than it is, if this
were the case. Unfortunately the supposition
is not true.