1.1. Sum of Countably Many Non-negative Terms#

Definition 1.1

Let \(A\) be a family of countably infinitely many non-negative terms in extended real numbers. Formally,

\[\begin{align*} A = \set{a_\alpha \in \overline{\R}_{\geq 0} }{\alpha \in \Lambda}\end{align*}\]

where \(\Lambda\) is a countably infinite index set. Then the sum of terms in \(A\) is defined by the sum of the series

\[\begin{align*}\sum_{n=1}^\infty a_{\sigma(n)}\end{align*}\]

where \(\sigma: \N^\ast \to \Lambda\) is a bijection.

Definition 1.2

Let \(\{a_n\}\) be a sequence, and \(\sigma: \N^\ast \to \N^\ast\) a bijection. Let \(\hat{a}_n\) be given by

\[\begin{align*}\hat{a}_n := a_{\sigma(n)}, \quad n \in\N^\ast\end{align*}\]

Then sequence \(\sigma\) is said to be an rearrangement of \(\{a_n\}\) into sequence \(\{\hat{a}_n\}\).

Theorem 1.1

Let \(\{a_n\}\) be a sequence of complex numbers, and \(\sigma\) an rearrangement. If the series \(\sum a_n\) converges absolutely to sum \(s\), then \(\sum a_{\sigma(n)}\) also converges absolutely to \(s\).

Corollary 1.1

Let \(\{a_n\}\) be a sequence consisting of non-negative terms in extended real numbers, possibly including infinity, and \(\sigma\) be an rearrangement. Then

(1.2)#\[\begin{align}\sum_{n=1}^\infty a_{\sigma(n)} = \sum_{n=1}^\infty a_n \end{align}\]

Note

Equation (1.2) should be understood as a compact expression containing the following two meanings.

  • ➀ If \(\sum a_n\) converges to a (non-negative) sum \(s\), then \(\sum a_{\sigma(n)}\) also converges to \(s\).

  • ➁ If \(\sum a_n\) diverges to infinity (either \(a_n\)’s are all finite numbers and the series diverges to infinity or there exists \(\infty\) among \(a_n\)’s), then \(\sum a_{\sigma(n)}\) also diverges to infinity.

Definition 1.3

A double sequence is in general any function whose domain is \(\N^\ast \times \N^\ast\). In this book, we are particularly interested in the function whose values are complex numbers, or non-negative extended real numbers. The double sequence is denoted by \(\{a_{m,n}\}\) where the function variables \(m, n \in \N^\ast\) are referred to as indices, and the function value \(a_{m,n}\) is called term.

Note

Concerning the usual notation for a function in above definition, the double sequence should be written as \(f(m, n)\) where \(m, n \in \N^\ast\). However, we adopt the notation \(a_{m,n}\) to emphasize that the function values, or terms, are indexed by two natural numbers.

Definition 1.4

Let \(\{a_{m,n}\}\) be a double sequence, and \(\{s_{p,q}\}\) be another double sequence defined by

\[\begin{align*} s_{p,q} = \sum_{m=1}^p \sum_{n=1}^q a_{m,n}\end{align*}\]

The pair \((\{a_{m,n}\}, \{s_{p,q}\})\) is called the double series, and is denoted by \(\sum_{m,n} a_{m,n}\). The sequence \(\{s_{p,q}\}\) is referred to as the \((p,q)\)-th partial sum of \(\sum_{m,n} a_{m,n}\).