Double Series of Real Numbers

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:

\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:

\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:

\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.

Unless otherwise stated, the content of this page is licensed under Creative Commons Attribution-ShareAlike 3.0 License