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.

Fourth Chapter.
Uniform convergence, Differentiation and Integration of an
infinite series.
127. In the General Theorems concerning series of powers,
(~ 44. IV), it was indicated that the proof of the continuity of a
function expressed by a series of powers, as well as the rule for
its differentiation, is based on a definite property of these series,
namely on their uniform or equable convergence.
We are now going to discuss this conception more closely for
any arbitrary infinite series which converges for a real interval.
Let the infinite series:
f (x) + f2(x) +- f3(x) + * - + f (x) + f+l(x) + -
be convergent in the interval from a to b; let its sum be denoted
by F(x). The functions fn(x) can be continued unrestrictedly according
to some law, and we assume that however many of them are formed
they are all continuous in the assigned interval. This hypothesis
is to be maintained in all the following theorems.
The convergence of the infinite series requires, that, for any
number 6 however small, it shall be possible to find a place n in the
series, such that every remainder:,Rn =f fn(x) + fn+l (X) +.,,n+ =fn, -(x) + fn+2 (X) +,
BRn+k fn —+(X) +.-.
from Rn onwards shall be smaller in amount than 6 (~ 39).
The infinite series is said to be uniformly or equably convergent in
the entire interval without exception, when this criterion of convergence
is satisfied by the same n for any given 6, zwhile x passes through
all values from a -to b; we need not therefore take a different value
for n until another value is assigned to 6.*)
A sufficient, although not a necessary, criterion of uniform
convergence is presented in the theorem: When the series of the
numerically greatest values assumed by the terms of the infinite
*) Heine: Ueber trigonometrische Reihen. Crelle's Journal, Vol. 71.