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[parent] real part series and imaginary part series (Theorem)

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
$\displaystyle a_1+a_2+a_3+\ldots\;\;\;$and$\displaystyle \;\;\;b_1+b_2+b_3+\ldots$
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

$\displaystyle S_n = z_1+\ldots+z_n = (a_1+ib_1)+\ldots+(a_n+ib_n) = (a_1+\ldots+a_n)+i(b_1+\ldots+b_n) := A_n+iB_n$
( $ 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
$\displaystyle \vert(A_n-A)+i(B_n-B)\vert = \vert(A_n+iB_n)-(A+iB)\vert < \varepsilon.$
But a complex number is always absolutely at least equal to the real part (see the inequalities in modulus of complex number), and therefore $ \vert A_n-A\vert \leqq \vert(A_n-A)+i(B_n-B)\vert < \varepsilon$, similarly $ \vert B_n-B\vert \leqq \vert(A_n-A)+i(B_n-B)\vert < \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
$\displaystyle a_1+a_2+a_3+\ldots = A\;\;\;$and$\displaystyle \;\;\;b_1+b_2+b_3+\ldots = B,$
Q.E.D. The converse part is straightforward.

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

$\displaystyle \vert z_1\vert+\vert z_2\vert+\vert z_3\vert+\ldots$
converges if and only if the series
$\displaystyle a_1+a_2+a_3+\ldots\;\;\;$and$\displaystyle \;\;\;b_1+b_2+b_3+\ldots$
converge absolutely.

Proof. Use the inequalities

$\displaystyle 0 \leqq \vert a_n\vert \leqq \vert z_n\vert,\quad 0 \leqq \vert b_n\vert \leqq \vert z_n\vert$
and
$\displaystyle 0 \leqq \vert z_n\vert \leqq \vert a_n\vert+\vert b_n\vert$
for using the comparison test.

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

$\displaystyle \left\vert\sum_{n=1}^\infty z_n\right\vert \leqq \sum_{n=1}^\infty\vert z_n\vert.$

Proof. By theorem 2, the convergence of $ \sum\vert z_n\vert$ 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

$\displaystyle \left\vert\sum_{j=1}^n z_j\right\vert \leqq \sum_{j=1}^n\vert z_j\vert,$
then we must have the asserted limit inequality, too.



"real part series and imaginary part series" is owned by pahio.
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See Also: sum of series, modulus of complex number, absolute convergence theorem

Keywords:  real part, imaginary part

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Cross-references: limit, triangle inequality, implies, comparison test, modulus of complex number, inequalities, complex number, integer, partial sum, number, positive, proof, converse, real, sum, converges, imaginary parts, real parts, series
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This is version 7 of real part series and imaginary part series, born on 2007-08-11, modified 2007-08-12.
Object id is 9854, canonical name is RealPartSeriesAndImaginaryPartSeries.
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Classification:
AMS MSC40-00 (Sequences, series, summability :: General reference works )

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