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

206 INTRODUCTION TO MATHEMATICS
values 1, 2, 3, etc., in succession, we get the
series
fl(zx,f2(x),f(x,..., f(x),...
Such a series may be convergent for some
values of x and divergent for others. It is,
in fact, rather rare to find a series involving a
variable x which is convergent for all values
of x,-at least in any particular instance it is
very unsafe to assume that this is the case.
For example, let us examine the simplest of
all instances, namely, the "geometrical"
series
1,xyx 2,.3.. n
X, Xe X 3......
The sum of n terms is given by
Sn=1+X+X2+X3 +... +x.
Now multiply both sides by x and we get
xSn =-+x2+3+x4+... +xn+n+l
Now subtract the last line from the upper
line and we get
Sn(l - x) = Sn- xs, = 1 - n+l,
and hence (if x be not equal to 1)
i - Xn+l 1 xn+l
Sn 1-X _T^X T^C
1-x 1-x 1-x
Now if x be numerically less than 1, for suffiXn
ciently large values of n, _ is always less
j1 x