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

208 INTRODUCTION TO MATHEMATICS
within the interval we have satisfied the
desired standard of approximation?
Sometimes we can and sometimes we cannot do this for each value of k. When we
can, the series is called uniformly convergent
throughout the interval, and when we cannot
do so, the series is called non-uniformly convergent throughout the interval. It makes
a great difference to the properties of a series
whether it is or is not uniformly convergent
through an interval. Let us illustrate the
matter by the simplest example and the
simplest numbers.
Consider the geometric series
1+X+X2+X3+... xn+.
-It is convergent throughout the interval
-1 to +1, excluding the end values x= ==1.
But it is not uniformly convergent throughout this interval. For if Sn(x) be the sum of
n terms, we have proved that the difference
1, Xn+
between Sn(x) and the limit 1 is 1-
Now suppose n be any given number of terms,
say 20, and let k be any assigned standard
of approximation, say.001. Then, by taking
x near enough to +1 or near enough to -1,
x21
we can make the numerical value of - to
1-x
be greater than *001. Thus 20 terms will