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

SERIES 209
not do over the whole interval, though it is
more than enough over some parts of it.
The same reasoning can be applied whatever other number we take instead of 20,
and whatever standard of approximation instead of.001. Hence the geometric series
1l+x+-x2+ 3+... +n+... is non-uniformly convergent over its whole interval of
convergence -1 to +1. But if we take any
smaller interval lying at both ends within the
interval -1 to +1, the geometric series is
uniformly convergent within it. For example, take the interval 0 to + 1. Then any
xn+l
value for n which makes - numerically
less than k at these limits for x also serves
for all values of x between these limits, since
Zn+l
it so happens that - diminishes in numerical value as x diminishes in numerical value.
For example, take k=.001; then, putting
x = -, we find:
xn+l 1 \2
for n=l, =0111
forn= 1 -X 1 =. —00111...,
for n=e, 1 -
I - x - wlo
yn+1 (J-Y
for n =$3, 1 - 1-=~T -6~.000111...
1-z 1-0l - o
Thus three terms will do for the whole in