Absolute Convergence of Double Series

Subjects: Real Analysis
Links: Double Series, Absolute Convergence Test and Properties

Definition: The double series $\sum _{n,m=1}^{\mathrm{\infty }}{z}_{n,m}$ is said to be absolutely convergent if $\sum _{n,m=1}^{\mathrm{\infty }}|z_{n,m}|$ converges. The iterated series $\sum _{m=1}^{\mathrm{\infty }}\left(\sum _{n=1}^{\mathrm{\infty }}{z}_{n,m}\right)$ is said to be absolutely convergent if $\sum _{m=1}^{\mathrm{\infty }}\left|\sum _{n=1}^{\mathrm{\infty }}{z}_{n,m}\right|$ converges.

Theorem: Every absolutely convergent double series is convergent.

Theorem: $\mathrm{\forall }n,m\in \mathbb{N}(z_{n,m}\in \mathbb{C})$, and $\varphi :\mathbb{N}\to {\mathbb{N}}^2$ be a bijection. If any of the following three sums:

$\sum _{n=1}^{\mathrm{\infty }}\sum _{m=1}^{\mathrm{\infty }}|z_{n,m}|$, $\sum _{m=1}^{\mathrm{\infty }}\sum _{n=1}^{\mathrm{\infty }}|z_{n,m}|$, $\sum _{k=1}^{\mathrm{\infty }}|z_{\varphi (k)}|$ is finite, then all of the following series:

1. $\sum _{n=1}^{\mathrm{\infty }}{z}_{n,m}$, for $m\in \mathbb{N}$
2. $\sum _{m=1}^{\mathrm{\infty }}{z}_{n,m}$, for $n\in \mathbb{N}$
3. $\sum _{n=1}^{\mathrm{\infty }}\sum _{m=1}^{\mathrm{\infty }}{z}_{n,m}$, $\sum _{m=1}^{\mathrm{\infty }}\sum _{n=1}^{\mathrm{\infty }}{z}_{n,m}$, $\sum _{k=1}^{\mathrm{\infty }}{z}_{\varphi (k)}$

are absolutely convergent and the three series in $(3)$ have the same sum

Theorem: $\mathrm{\forall }n,m\in \mathbb{N}(z_{n,m}\in \mathbb{C})$, and let $\varphi :\mathbb{N}\to {\mathbb{N}}^2$ be a bijection. Then,

1. $\sum _{k=1}^{\mathrm{\infty }}{z}_{\varphi (k)}$ converges absolutely iff $\sum _{n,m=1}^{\mathrm{\infty }}{z}_{n,m}$ converges absolutely.
2. If $\sum _{n,m=1}^{\mathrm{\infty }}{z}_{n,m}$ converges absolutely to the sum $s$, then $\sum _{k=1}^{\mathrm{\infty }}{z}_{\varphi (k)}=s$.

Theorem: $\mathrm{\forall }n,m\in \mathbb{N}(z_{n,m}\in \mathbb{C})$. If any of the following three sums:

$\sum _{n=1}^{\mathrm{\infty }}\sum _{m=1}^{\mathrm{\infty }}|z_{n,m}|$, $\sum _{m=1}^{\mathrm{\infty }}\sum _{n=1}^{\mathrm{\infty }}|z_{n,m}|$, $\sum _{n,m=1}^{\mathrm{\infty }}|z_{n,m}|$ is finite, then all of the following series:

1. $\sum _{n=1}^{\mathrm{\infty }}{z}_{n,m}$, for $m\in \mathbb{N}$
2. $\sum _{m=1}^{\mathrm{\infty }}{z}_{n,m}$, for $n\in \mathbb{N}$
3. $\sum _{n=1}^{\mathrm{\infty }}\sum _{m=1}^{\mathrm{\infty }}{z}_{n,m}$, $\sum _{m=1}^{\mathrm{\infty }}\sum _{n=1}^{\mathrm{\infty }}{z}_{n,m}$, $\sum _{n,m=1}^{\mathrm{\infty }}{z}_{n,m}$

are absolutely convergent and the three series in $(3)$ have the same sum

Theorem: Let $\sum _{n=1}^{\mathrm{\infty }}{a}_n$, and $\sum _{m=1}^{\mathrm{\infty }}{b}_m$ be absolutely series of complex numbers with sums of $a$ and $b$, respectively. Let $z_{n,m}$ be a double sequence defined as:

Then the double series $\sum _{n,m=1}^{\mathrm{\infty }}{z}_{n,m}=ab$.