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.

224

Differentiation of an infinite series.

B3k. III. h. IV.

function (D shall differ inappreciably from the quotient of differences;
there must therefore also be a point at which:
F(x + )-F(x) - = (x + OAx);
Ax
and since 0) as a uniformly convergent series signifies a continuous
function, we shall have for Ax = 0:
F (x) = d (x).
Series, to which these criteria do not apply, cannot be differentiated
except by attempting to sum directly the infinite series for the quotient
of differences -F(x A) - () and then passing to the limit for Ax.
Ax
Examples.
1) It was seen in ~ 47 that the infinite series:
x2 x3 x4
x - +3- -+ ~
has for - 1 < x < + 1 the value:
l(1 + x).
This series converges uniformly. The series formed of its derived terms:
1 - X -- x2 - X3 + ~. ~
converges uniformly for - 1 < x < + 1, and is thus a continuous
1
function. It expresses the derived function -; but this connexion
does not hold for x = 1, although the differential quotient of the
logarithm has the determinate value I.
2) It is shown in the Theory of Trigonometric Series, that the
signification of the infinite series:
3- -- 52
when 0 < x < - is F(x) ==x; and when 2- <x_, is F(x) -- x;
it is uniformly convergent. Further, we have:
F'() _ 4 (Cos X cosx cosr 5x
except for x = -- for which the derived series is discontinuous and
expresses the value zero, while the progressive differential quotient of
F(x) is --- and the regressive + 1.
3) The infinite series whose general term is:
1 1 /(2nq- 12x -- 1)
fn(X) = 2n (2 X + 1) - 2(n + 1)2 +
is uniformly convergent for all finite values of x, since:
X (x) = 1 1 (n2 + 1).
2n x