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.

222

Differentiation of an infinite series.

Bk. III. ch. IV.

129. We come now in the second place to investigate, under
what condition the differential quotient of an infinite series is expressed
by the series formed of the differential quotients of its several terms.
Here we assume that all the functions f(x) can be differentiated and
that their derived functions are continuous; but, moreover since the
infinite series:
fl'(x) + f2'(x) + fa3(x) +. * f'() +-
cannot possibly be convergent unless Lim f,'(x) vanish for n = co, )
our investigation must be based on the hypothesis that it does vanish.
Example. It is shown in the Theory of Trigonometric Series,
that for - t < x < +- r the infinite series:
sin 2x sin 3x sin 4x
sin x - - -+- + -
expresses the value I x; that is to say, a function whose derived is.
But this value is not presented by the series got by differentiating
the several terms:
cos x - cos 2x + cos 3x - cos 4x + * v
which does not even converge, but is completely indeterminate, because:
Lim fn,(x) =- - Lim cos n x
assumes for n = o all possible values between - 1 and +- 1.
In order to determine the differential quotient of the function
F(x) at a point in which I'(x) is continuous, let us first form the
quotient of differences:
converges, for, the remainder:
=n () 2 2 +X 1
is zero for n =-, and the sum of the series is the continuous function:
F(x) = Zx
X2 + 1
But in the neighbourhood of the point x = 0 this series converges unequably;
for, the function Rn (x) has maximum and minimum values + I for x=+- -
Therefore near zero no interval can be assigned however small, within which
the amounts of all remainders after a certain one continue smaller than an
arbitrarily small number.
i) Lim f '(x) denotes that fn'(x) is first formed and then n put = oo. This
d
is of course to be distinguished from d- Lim fn(x) in which we first put n = co
and then differentiate.
If /(x) ==n nx, Lim fn'(x) Lim cos nx is completely indeterminate,
while on the other hand:
d. sin nzx
Lim -- = 0.
dxr n