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.

76

Calculation of functions by infinite series.

Bk. 1. ch. V1III.

V. The infinite series of powers is differentiated, by forming the
series of differential quotients of its several terms.
The series
a, + 2 a, x +- a3 x2 +... n ann-1 + * *.
derived from
f ( a) -- a. + a x + 2 x2 +.* Axn +...
certainly converges for all values of x which lie within the interval
of convergence of the original series. For, the interval of the derived
series is, according to the criterion, determined by
(n+1)a~x a an
abs Lim +- 1 a 1, or abs x < abs Lim
n a. a n+, +
Now since Lim -+ Lim (l- 1 -+-) becomes =1, for n o, it
n 4-1 --
a
follows that abs x < abs Lim
an+l
Now in order to determine the differential quotient of the continuous function f (x), let us first form the quotient of differences,
doing so ex. gr. regressively, in order when possible to take account
also of the upper limit of its convergency:
f(x - Ax) -- f (x) _ p (x - A x - p (x) P ( -Ax) - (x)
- Ax -Ax + -Ax
For any finite Ax however small, this continuous expression in Ax
has a determinate finite value.
If we denote the infinite series a1 +- 22 x - + - + nan x-'1 +
by X (x), its remainder by Rn(x), this equation takes the form
/'(x — A x -f(x) Ax) (x — ) + q A) - p(x)
— Ax -— = ( - x)- A-(- oaAx
Retaining the value of Ax, when we increase n arbitrarily the value of
0 changes on the right side. But as the remainder of a series of
powers has the property, that after some determinate n, RX (x) becomes
arbitrarily small for all values between x and x-Ax, then because as
n increases the last quotient can also be made arbitrarily small, it
follows that
f (x -Ax) -- f (X)
-_Az c + 2 a, (x - O x) + a3 a, (x - e Ax)2 +..
For because the continuous function (x) comes arbitrarily near
the quotient of differences in the interval from x to x -- Ax, there
must also be a point (compare ~ 17) where the two are equal. Now
the differential quotient f' (x) arises from the quotient of differences by
continuous transition for Ax = 0. But as long as what is on the
right side converges, it is by Theorem IV a continuous function of
the variable x - OAx, therefore we have for Ax = 0: