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.

122

Complex series.

Bk. II. ch. II.

require of convergence is that of absolute or unconditional convergence; but when their terms have different signs, ex. gr. are alternately positive and negative, a special property is enunciated. (Cf.
~ 44, III.)
If a complex series converges absolutely, the series of its moducli
Y/Uj -I — Vj + i/7ai ~ v12 + Y2 4+ V2 -i — 1/u2 +4 v2 +"/t(o + vo2 + |'12/u + v,2 + /,t,2 + v22 + ~ ~ ~ In + v + ~ ~.
also converges.
If we denote the terms u +- iv separated according to the signs
of u and v so that
1) (( + ) — iv) = ( 2( (1) + iV(')) + ( 2 ((2) - iV(2) ) + 2(-u -u(3) + iV(3))
+ 2:( — ') - i(4)),
then on the hypothesis of absolute convergence, each of the series:
2) 2 t(,1), 2;(2), ut"(3) 2tL(4), ZV(O)) 2V(2), 2V(3) 2 V(4)
converges, and therefore by the theorem of addition the series:
3) (1((1) + v(1)), _'(u(2) + V(2)), Z(u(3) + V(3)), 2(u4 + (4) ).
also converge. But now:
4) 1 (') + v()' > J/(tl())2 + (v('))2, u(2) + v(2) > J/(u(2))2 + (v(2))2
i1(3) + V(3) > 1/(u(3))'2 + (V(3))2, u(4) + V(4) > 1/(u(4))'2 _ (V())
Consequently:
5) 1/J/(U( '))'-(v())j2, 2l'/(u(2)))2 + (V(2))2 21/((-))2+(v(3))2
v('((4))2 + (v(4))2
must have finite values, and therefore the sum of these four series
21/u2 + v2 is likewise a finite quantity.
The converse proposition is also true; for since the modulus of a
sum is not greater than the sum of the moduli of its summands,
when the sums 5) have finite limiting values, the moduli of the sums
2(ut(1) -+- V(1)), 2((u(2) - iV(2)), 2(- t(3) -- V(3)), 2(- u(4) - iv(4))
must also be finite; this requires that the sums 2) should converge.
The necessary and sufficient condition for the absolute convergence
of a complex series is therefore the convergence of the series of moduli
belonging to it.*)
An absolutely convergent series tends to the same finite limiting
value, and is therefore said to have the same sum, even when the
arrangement of its terms is changed according to any law.
Let U,n + iVn be the sum of the terms
(Uo + ivo) + (u, + iv,) + * * (un + ivy),
and Lim (U,, + iV,) - U + iV; further let:
) Cauchy Cours d'Analyse algbrique.
S) Cauchy: Cours d'Analyse algebrique.