Double Series of Real Numbers

Recall that if $(a_n)_{n=1}^{\infty}$ is a sequence of real numbers then the corresponding series $\displaystyle{\sum_{n=1}^{\infty} a_n}$ is the sum of all terms in the sequence $(a_n)_{n=1}^{\infty}$.

We also defined a corresponding sequence of partial sums to this series denoted $(s_n)_{n=1}^{\infty}$ where for all $n \in \mathbb{N}$ we have that:

(1)

\begin{align} \quad s_n = \sum_{k=1}^{n} a_k = a_1 + a_2 + ... + a_n \end{align}

We will now extend these concepts to double sequences of real numbers.

Definition: Let $(a_{mn})_{m,n=1}^{\infty}$ be a double sequence of real numbers. Then the corresponding Double Series is denoted $\displaystyle{\sum_{m,n=1}^{\infty} a_{mn}}$. The corresponding Double Sequence of Partial Sums denoted $(s_{mn})_{m,n=1}^{\infty}$ is defined for each $m, n \in \mathbb{N}$ by $\displaystyle{s_{mn} = \sum_{j=1}^{m} \left ( \sum_{k=1}^{n} a_{jk} \right )}$ called the $(m,n)^{\mathrm{th}}$ Partial Sum (or Double Partial Sum). We say that the double series $\displaystyle{\sum_{m,n=1}^{\infty} a_{mn}}$ converges to $A \in \mathbb{R}$ if the corresponding double sequence of partial sums $(s_{mn})_{m,n=1}^{\infty}$ Converges to $A$, and if $(s_{mn})_{m,n=1}^{\infty}$ and diverges if $(s_{mn})_{m,n=1}^{\infty}$ does not converge for any $A \in \mathbb{R}$.

We can visualize the $(m,n)^{\mathrm{th}}$ partial sum of the double series $\displaystyle{\sum_{m,n=1}^{\infty} a_{mn}}$ as the sum of all terms contained in the $m$ by $n$ box when the terms of the double sequence $(a_{mn})_{m,n=1}^{\infty}$ are arranged in an array:

For example, consider the following double series:

(2)

\begin{align} \quad \sum_{m,n=1}^{\infty} \frac{1}{m+n} \end{align}

If $(s_{mn})_{m,n=1}^{\infty}$ is the double sequence of partial sums for this double series then the $(2,3)^{\mathrm{th}}$ partial sum is:

(3)

\begin{align} \quad s_{2,3} &= \left ( \frac{1}{1 + 1} + \frac{1}{1 + 2} + \frac{1}{1 + 3} \right ) + \left ( \frac{1}{2 + 1} + \frac{1}{2 + 2} + \frac{1}{2 + 3} \right ) \\ \quad s_{2,3} &= \left ( \frac{1}{2} + \frac{1}{3} + \frac{1}{4} \right ) + \left ( \frac{1}{3} + \frac{1}{4} + \frac{1}{5} \right ) \\ \quad s_{2,3} &= 1.8666... \end{align}

Now also recall that in general, any double sequence of real numbers $(a_{mn})_{m,n=1}^{\infty}$ has two iterated limits, $\displaystyle{\lim_{m \to \infty} \left ( \lim_{n \to \infty} a_{mn} \right )}$ and $\displaystyle{\lim_{n \to \infty} \left ( \lim_{m \to \infty} a_{mn} \right )}$ that may or may not exist. We can define two iterated double sums for a double series as well!

Definition: Let $\displaystyle{\sum_{m,n=1}^{\infty} a_{mn}}$ be a double series of real numbers. Then the corresponding Iterated Double Sums are $\displaystyle{\sum_{m=1}^{\infty} \left ( \sum_{n=1}^{\infty} a_{mn} \right )}$ and $\displaystyle{\sum_{n=1}^{\infty} \left ( \sum_{m=1}^{\infty} a_{mn} \right )}$.

Once again, it is important to note that in general, these two iterated double sums may not be equal.