convergence of complex term series

A series

ν=1cν=c1+c2+c3+ (1)

with complex terms

cν=aν+ibν  (aν,bνν)

is convergent iff the sequence of its partial sums converges to a complex numberMathworldPlanetmathPlanetmath.

Theorem 1.  The series (1) converges iff the series

ν=1aνandν=1bν (2)

formed by real parts and the imaginary parts of its terms both are convergent.

Proof.  Let  ε>0.  Denote


If the series (2) are convergent with sums S and T, then there is a number N such that




i.e. the series (1) converges to S+iT.  If, conversely, (1) converges to a complex number




and consequently,  limnsn=s  and  limntn=t, i.e. the series (2) are convergent with sums the real numbers s and t.

Theorem 2.  The series (1) converges absolutely iff the series (2) both converge absolutely.

Proof.  The absolute convergence of (1) means that the series


converges.  But since  |cν|2=|aν|2+|bν|2,  we have


From these inequalities we can infer the assertion of the theorem 2.


  • 1 R. Nevanlinna & V. Paatero: Funktioteoria.  Kustannusosakeyhtiö Otava. Helsinki (1963).
Title convergence of complex term series
Canonical name ConvergenceOfComplexTermSeries
Date of creation 2014-10-31 19:04:59
Last modified on 2014-10-31 19:04:59
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 8
Author pahio (2872)
Entry type Theorem
Classification msc 30A99
Classification msc 40A05
Related topic OrderOfFactorsInInfiniteProduct