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.

72

Calculation of functions by infinite series.

Bk. I ch. VIII.

are to be considered as definitions of these functions, and from them
all their properties already employed must directly follow.
44. To demonstrate this, we must show independently of previous
considerations that the defining series converge and are continuous
functions of x. For this purpose we prove the following
General theorems concerning series of powers.*)
I. If the coefficients ao, act.... an... in any series of powers
f (x) = oa0 + a1x 2 x2 + * ' a.nX +- * *
are all of like sign, and for a definite positive value X its terms after
some certain one decrease and converge to zero, so that the quotient of
a term by the preceding one is less than unity, and for n- == is at
most equal to unity, then the series converges for all positive values of x
which are smaller than X.
The quotient a-n+ X after some definite place in the series being
less than or at most equal to unity, if we take x < X, a proper
fraction a can be assigned, such that
a -n+l X an+2 X a<n+3 X
< < a, < a 3. *,
an an-+1 an-+2
therefore
an+i x < a( an an+2 X < a2 an,.. t (tn+k < ak n...
hence
anXn Xn+ (an+ Xn+1 *.. +a C Xnk + < al 1 + a a * *. +ak.. *.}
or:
an x + -1X + x a. n+ *-+xk X + *.. < an X" 1-_ (
The quantity on the right is finite and positive and we can give
n such a value as will make it less than any assignable quantity.
Our statement is therefore proved.
For = 1 this reasoning applies no longer; so that we have
always specially to investigate whether a series continues to converge,
in case the quotient of two consecutive terms tends to the limiting
value unity.
If on the other hand for a value x, the ratio of a term to the
preceding one, from some initial place in the series, is always greater
than I, even though it may be = 1 for n = co, the series will have no
determinate finite sum for this or any greater value of x, i. e. it will
diverge, because the succeeding terms increase, and therefore also the
remainder of the series does not converge to zero.
*) Abel (1802-1829): Recherches sur la serie I + -- x - 1 --- + etc.,
Oeuvres 1, p. 219 (1881). Crelle J. Vol. 1, p. 311 (1826).