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.

.146

Functions of a complex variable.

Bk. II. ch. II.

the original series 1) is convergent; for, B its radius of convergence
is determined from the inequality ~ 83:
B Lim +- At- < 1, or R < Lim A, since Lim +-.
n A A-+1 nO= ic
Similarly we obtain by continued differentiation of the several terms
the following series that all converge within the same circle:
4) 2ca + 3.2a3- -+.*. e n - 1) an-2... ( = ()
3.2a3 + 4. 3. 2a4z + n(n - 1) (n- 2)an^z-3... f3(z)
Introducing this notation we obtain for series 2) arranged by powers
of h the value:
5) h h. hn
5) f ( + h) = f() +- 1 f (,) + 2 f2 (() +.. () + '..
where Lim h- fn () is certainly zero, because the sum of the moduli of
~n ft (z) is smaller than, the sum of the moduli of the absolutely
convergent series of powers:
an (z + h)n+ an+i( + h)+rl + a+n2 ( + by)n+2 +... etc..
The convergency of this new series 5) is therefore a circle with its
centre at the point h = 0, i. e. at the point 2, and its radius H at
least equal to: B - mod [z]; for, as the circle, whose radius is equal
to this, touches the inside of the original one, all its points lie within
that circle and for them series 2) converges absolutely, therefore series 5)
derived from it by arranging its terms differently also converges absolutely.
We have still to convince ourselves directly, that it is allowable
to reason thus from the absolutely convergent series 2), for this might
seem doubtful, since in the new arrangement of terms each coefficient
requires the summation of an infinite series. Let us therefore examine
whether n can be chosen so as to make the difference between the first
gn -+ terms of series 2) and of series 5) arbitrarily small, always
assuming the absolute convergence of the former series. Putting:
f' () a — 0 + a, + a2 2 + * * an + Qn,
ft (=) = ca + 2 a, +... nan - + ',,
~2 (~ ) 2 a2 + 3.2 a3 + * * - i (n -1) a.n - 2 + o,~
fn(i) == 1. Con + n (n),
the difference between the sums of their first n + 1 terms is:
+ 1 Qn + -- 2 n + - * Z(a)
h, 12 hn
Now since series 2) converges absolutely, we can always choose n, such
that for every value of k: