An introduction to the study of the elements of the differential and integral calculus. From the German of the late Axel Harnack, With the permission of the author.

74

Calculation of functions by infinite series. Bk. I ch VIII.

a greater value it becomes greater than one; the terms of the series
then do not decrease in amount but increase.
IV. Every series of powers is a continuous function of the variable,
withizn the interval in which it absolutely converges.
Let f (x) signify the value of the infinite series
a0 + ai, + a 2 x2 * Ctn X + * *.
for which, since x is to be a value within its convergency, we have
Lim [ +1 xl <1,
it is required to show that Lim [f(x + ) - f (x)] = 0 for - 0.
Putting:
a0 + a1 x + a 2 X2-. an-* xn-~ = T (x)
aC xn + Cla,+ xn+1+n2 +a+ X+2... ' (x),
then as in Theorem I
abs ip(x) is < abs anx.n x 1 -, (0 < a < 1).
Merely by the selection of a lower limit for n, we can thus make i (x)
as well as (x + 6) and therefore also the amount of their difference
t (x + 6) -- (x) less than a quantity E however small, because the
term an xn becomes arbitrarily small as n increases. When X denotes
the greatest value of the interval for x, we must choose n so that
ac < -. ) Accordingly
Xn
f(x 4- 6 - f (x) = p (x +_ ) - (x) + E.
Now since p (x) denotes a rational integer function of x, which as
already seen ~ 19 is continuous, the difference f(x ~- 6) - f(x) becomes
smaller than an arbitrarily small quantity as d decreases, i.e. f (x)
is a continuous function.
The Theorem also holds when the series converges at the limits
of the interval of convergence for X, conditionally or unconditionally:
that, for d == 0
Lim f(X - 6) = (X).
For we have here:
*) In consequence of this property, that for the same t, both ) (x) and
9 (x + d) become less than a, series of powers are said to be convergent in
equal degree or uniformly. Abel was the first (loc. cit. Oeuvres I, p. 225)
to point out, that continuity of the series does not of itself follow from the
continuity of the terms of the series. Uniform convergence teaches also, that
the infinite series in its entire convergency can be replaced quam proxime by the
same rational integer function. The function expressed by the series of powers
is styled therefore, after Weierstrass, one which has the character of a rational
integer function.