real part series and imaginary part series

Theorem 1. Given the series

z1+z2+z3+ (1)

with the real partsMathworldPlanetmath of its terms  zn=an  and the imaginary parts of its terms  zn=bn  (n=1, 2, 3,). If the series (1) converges and its sum is A+iB, where A and B are real, then also the series


converge and their sums are A and B, respectively. The converse is valid as well.

Proof. Let ε be an arbitrary positive number. Denote the partial sum of (1) by


(n=1, 2, 3,). When (1) converges to the sum A+iB, then there is a number nε such that  for any integer  n>nε  we have


But a complex numberMathworldPlanetmathPlanetmath is always absolutely at least equal to the real part (see the inequalitiesMathworldPlanetmath in modulus of complex number), and therefore  |An-A||(An-A)+i(Bn-B)|<ε, similarly  |Bn-B||(An-A)+i(Bn-B)|<ε  as soon as  n>nε.  Hence,  AnA  and  BnB  as  n.  This means the convergences


Q.E.D. The converse part is straightforward.

Theorem 2. Notations same as in the preceding theorem. The series


converges if and only if the series


converge absolutely (

Proof. Use the inequalities




for using the comparison testMathworldPlanetmath.

Theorem 3. If the series n=1|zn| converges, then also the series n=1zn converges and we have


Proof. By theorem 2, the convergence of |zn| implies the convergence of an and bn, which, by theorem 1, in turn imply the convergence of zn . Since for every n the triangle inequalityMathworldMathworldPlanetmath guarantees the inequality


then we must have the asserted limit inequality, too.

Title real part series and imaginary part series
Canonical name RealPartSeriesAndImaginaryPartSeries
Date of creation 2013-03-22 17:28:08
Last modified on 2013-03-22 17:28:08
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 10
Author pahio (2872)
Entry type Theorem
Classification msc 40-00
Related topic SumOfSeries
Related topic ModulusOfComplexNumber
Related topic AbsoluteConvergenceTheorem
Related topic RealAndImaginaryPartsOfContourIntegral