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

SERIES 207
than k, however k be chosen. Thus, if x be
less than 1, the series 1, x, x2,...,... is
convergent, and 1 is its limit. This statement is symbolized by
-= 1-+X+i2+.. +x-.., (x<l).
1-x
But if x is numerically greater than 1, or
numerically equal to 1, the series is divergent.
In other words, if x lie between -1 and +1,
the series is convergent; but if x be equal
to -1 or to +1, or if x lie outside the interval
-1 to + 1, then the series is divergent. Thus
the series is convergent at all "points"
within the interval - 1 to +1, exclusive of
the end-points.
At this stage of our enquiry another question arises. Suppose that the series
fi(x) +f(x) +f(x)+. - * +fn(x)+...
is convergent for all values of x lying within
the interval a to b, i.e. f(x) is convergent for
any value of x which is greater than a and
less than b. Also, suppose we want to be
sure that in approximating to the limit we
add together enough terms to come within
some standard of approximation k. Can we
always state some number of terms, say n,
such that, if we take n or more terms to
form the sum, then whatever value x has