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

205 INTRODUCTION TO MATHEMATICS
For example the series
111 1
l, 9,
2' 3' 4' n' *
is divergent. It is easy to see that this is
the case; for consider the sum of n terms
beginning at the (n+l)th term. These n
1 1 1 1
terms are,,...: there
n+l' n+2' n+3 2n
1
are n of them and - is the least among them.
Hence their sum is greater than n times
1.
i.e. is greater than - Now, without
altering the sum to infinity, if it exist, we
can add together neighbouring terms, and
obtain the series
1, +1, + + +, etc.,
that is, by what has been said above, a series
whose terms after the 2nd are greater than
those of the series,
1, 1, 1, ~, etc.,
where all the terms after the first are equal.
But this series is divergent. Hence the
original series is divergent.
This question of divergency shows how
careful we must be in arguing from the pro