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

210 INTRODUCTION TO MATHEMATICS
terval, though, of course, for some parts of
the interval it is more than is necessary.
Notice that, because 1 + x + x2 +...
+x n+... is convergent (though not uniformly) throughout the interval -1 to +1,
for each value of x in the interval some number of terms n can be found which will satisfy
a desired standard of approximation; but,
as we take x nearer and nearer to either end
value +1 or -1, larger and larger values of
n have to be employed.
It is curious that this important distinction
between uniform and non-uniform convergence was not discovered till 1847 by Stokesafterwards, Sir George Stokes-and later, independently in 1850 by Seidel, a German
mathematician.
The critical points, where non-uniform convergence comes in, are not necessarily at the
limits of the interval throughout which convergence holds. This is a speciality belonging
to the geometric series.
In the case of the geometric series 1+x
+ x2 +... + xn+..., a simple algebraic
expression 1 - can be given for its limit in
its interval of convergence. But this is not
always the case. Often we can prove a series
to be convergent within a certain interval,
though we know nothing more about its
limit except that it is the limit of the series.