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

a1+a2+a3+andb1+b2+b3+

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

Sn=z1++zn=(a1+ib1)++(an+ibn)=(a1++an)+i(b1++bn):=An+iBn

(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

|(An-A)+i(Bn-B)|=|(An+iBn)-(A+iB)|<ε.

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

a1+a2+a3+=Aandb1+b2+b3+=B,

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

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

|z1|+|z2|+|z3|+

converges if and only if the series

a1+a2+a3+andb1+b2+b3+

converge absolutely (http://planetmath.org/AbsoluteConvergence).

Proof. Use the inequalities

0|an||zn|,0|bn||zn|

and

0|zn||an|+|bn|

for using the comparison testMathworldPlanetmath.

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

|n=1zn|n=1|zn|.

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

|j=1nzj|j=1n|zj|,

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