real part series and imaginary part series

Theorem 1. Given the series

$\displaystyle z_{1}+z_{2}+z_{3}+\ldots$

(1)

with the real parts of its terms $\Re{z_{n}}=a_{n}$ and the imaginary parts of its terms $\Im{z_{n}}=b_{n}$ ( $n=1,\,2,\,3,\,\ldots$ ). If the series (1) converges and its sum is $A+iB$ , where $A$ and $B$ are real, then also the series

$a_{1}+a_{2}+a_{3}+\ldots\;\;\;\mbox{and}\;\;\;b_{1}+b_{2}+b_{3}+\ldots$

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

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

$S_{n}=z_{1}+\ldots+z_{n}=(a_{1}+ib_{1})+\ldots+(a_{n}+ib_{n})=(a_{1}+\ldots+a_% {n})+i(b_{1}+\ldots+b_{n}):=A_{n}+iB_{n}$

( $n=1,\,2,\,3,\,\ldots$ ). When (1) converges to the sum $A+iB$ , then there is a number $n_{\varepsilon}$ such that for any integer $n>n_{\varepsilon}$ we have

$|(A_{n}-A)+i(B_{n}-B)|=|(A_{n}+iB_{n})-(A+iB)|<\varepsilon.$

But a complex number is always absolutely at least equal to the real part (see the inequalities in modulus of complex number), and therefore $|A_{n}-A|\leqq|(A_{n}-A)+i(B_{n}-B)|<\varepsilon$ , similarly $|B_{n}-B|\leqq|(A_{n}-A)+i(B_{n}-B)|<\varepsilon$ as soon as $n>n_{\varepsilon}$ . Hence, $A_{n}\to A$ and $B_{n}\to B$ as $n\to\infty$ . This means the convergences

$a_{1}+a_{2}+a_{3}+\ldots=A\;\;\;\mbox{and}\;\;\;b_{1}+b_{2}+b_{3}+\ldots=B,$

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

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

$|z_{1}|+|z_{2}|+|z_{3}|+\ldots$

converges if and only if the series

$a_{1}+a_{2}+a_{3}+\ldots\;\;\;\mbox{and}\;\;\;b_{1}+b_{2}+b_{3}+\ldots$

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

Proof. Use the inequalities

$0\leqq|a_{n}|\leqq|z_{n}|,\quad 0\leqq|b_{n}|\leqq|z_{n}|$

and

$0\leqq|z_{n}|\leqq|a_{n}|+|b_{n}|$

for using the comparison test.

Theorem 3. If the series $\displaystyle\sum_{n=1}^{\infty}|z_{n}|$ converges, then also the series $\displaystyle\sum_{n=1}^{\infty}z_{n}$ converges and we have

$\left|\sum_{n=1}^{\infty}z_{n}\right|\leqq\sum_{n=1}^{\infty}|z_{n}|.$

Proof. By theorem 2, the convergence of $\sum|z_{n}|$ implies the convergence of $\sum a_{n}$ and $\sum b_{n}$ , which, by theorem 1, in turn imply the convergence of $\sum z_{n}$ . Since for every $n$ the triangle inequality guarantees the inequality

$\left|\sum_{j=1}^{n}z_{j}\right|\leqq\sum_{j=1}^{n}|z_{j}|,$

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