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.

124

Complex series,

Bk. II. ch. II.

of all the negative, both these series increasing beyond any finite amount;
let us form a series by first taking so many a terms that their sum is
greater than C, then joining on so many b terms that the sum becomes less than C, and so on. As this alternating arrangement is continued, the deviation from C will never amount to more than the
value of the term preceding the last change of sign; therefore since
the quantities a and b converge to zero, the value of the sum of the
series has the limit C.1 )
78. Multiplication of two infinite series.
If the moduli of the terms of the two complex series (~ 76):
1) Po + PA + P2 + 2 * Pn * and 2) q0 + + q, +q - q. n
(J2n =" Un + in l, qn = Un' + ivn')
likewise form convergent series:
3) 0 + 1 + +.+ e n + * ' ' 4) g0o' -+ 9 + c2 + Qn + '
(Q' -= 1//lC2+ V2 X Q -= tn2 + Vn2)
then the series
5) Poqo q+ (P0 q1 + P1 qo) + (p0 q2 + P1ql + P2 qO) + -*
*. ~ (o qn + Pi qn-i + * ~ qpkn- + * * - 7P qo) +..
is convergent and its sum is P. Q.
To prove this theorem*::) we require the Lemma:
If R and B' be the sums of the two series 3) and 4) that consist
only of positive terms, the series
6) Qoo' + (Qo0 1' + 1 Qo')+ (Qo + ' + 1,' + 92 o0') + *
* * (oQ en + 1 Qin-1 + * * ~ Qkn-k + * Qno') + *
is convergent and its sum = R. B'.
For, denoting the sum of the first n + 1 terms of series 3), 4)
and 6) respectively by Rn, Rn', Sn, it is plain, that Sn < Ban n'; and if
we call in the greatest integer in 4 n, that Sn > R,,nzR '. For, the
product (o + Q1 + * * * Qn) (o' + ' + * * * qn') contains more terms
than occur in S,, while all the terms of the product
(Q0 + Q1 + *- Q.) (Q0' + Q1' + '- Om')
occur in Sn, and in addition to them other positive quantities. It
follows from the inequality, which holds for every n however great,
an n' > Sn > IB'y Rm n
that Lim Sn = Lim Rn Rn =- Lim BRa', since as n is arbitrarily
increased Lim Rn - Lim?,,, Lim Ra' = Lim RB'. Forming now the
difference:
~) Dirichlet: loc. cit.; Riemann: Ueber die Darstellbarkeit einer Function
durch eine trigonometrische Reihe. Werke, p. 221.
*) Cauchy: Cours d'Analyse algebrique.