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.

~ 44.

Continuity of an infinite series.

75

g (X) = an X +- Cn+l Xn+l +,Z_,+2 Xn+2 +-.
u4 (X- 6) = (X Xn X ),+ X `+ + C n +
0P (X- a) -- a,,,,-~ —/ - a,+x
If the terms in X (X) are all of like sign, ex. gr. positive, we see
at once, that as < 1, I (X — 8 ) < ( (X), so that
a value can be given to n which will make g (X - 6) as well as
ip (X), smaller than any arbitrarily small quantity 8.
But if the terms in p (X) are different in sign, a special investigation is still required. This is based upon the following Lem n L a:
If to, tl, t,... t,,.. denote an infinite series of arbitrary quantities, and if the quantity
Pm = to + t + - * * t
for all values of mn be always algebraically less than a determinate
quantity G, but greater than g, then if %7, 1.. denote decreasing
positive quantities we have
g Eo < r - -0 t,, + l1 t, +... tL < G E.
Since
t -- P0, tl -I -- 2o0 t2 P 2 P- Pi etc.
therefore
r= oPo + l (2P - Po) + E2 (P2 —P.) + ' + + 8lm (Pml -- P1,-),
or
r == Po (EO -- E) + Pt (El -- E2) + Pm -1 (E1,m-l - mll) + Pm Em.
As the differences 8( - El 8, -,... are positive, the value of this
expression is less than
G (%0 - ~ + E -- 2,'. ~~ + E,) G EO
on the other hand it is greater than
9 (%o - 1l + e1 - E2 * * * -E + El,) = g.0
Applied to the present case, in which (X -~) (X X) denote
a series of decreasing positive quantities, it results from this Lemma
that the amount of
Q (X- 6) is less than (X 8) M,
where M represents the greatest numerical value in the series
a, Xn X + aan+ Xn +a,.., Xn -, Xon+ 1 - n+. X n+k X-, etc.
Since the series f(X) converges, a place an can be found in it from
which onwards the value of 31 is less than an arbitrarily small quantity E, whence what we stated follows.