Double Series of Real Numbers and Cauchy Products Review

Double Series of Real Numbers and Cauchy Products Review

We will now review some of the recent material regarding double series of real numbers and Cauchy products.

• On the Double Series of Real Numbers we said that if $(a_{mn})_{m,n=1}^{\infty}$ is a double sequence of real numbers then the corresponding Double Sequence of Partial Sums is the double sequence $(s_{mn})_{m,n=1}^{\infty}$ defined for all $m, n \in \mathbb{N}$ by:
(1)
\begin{align} \quad s_{mn} = \sum_{j=1}^{m} \left ( \sum_{k=1}^{n} a_{jk} \right ) \end{align}
• We said that the corresponding Double Series denoted $\displaystyle{\sum_{m,n=1}^{\infty} a_{mn}}$ (which denotes the sum of all terms in the double sequence $(a_{mn})_{m,n=1}^{\infty}$ Converges to $A \in \mathbb{R}$ if $(s_{mn})_{m,n=1}^{\infty}$ converges to $A$, and Diverges if it does not converge to any $A \in \mathbb{R}$.
• On the Absolute and Conditional Convergence of Double Series of Real Numbers page we said that a double series $\displaystyle{\sum_{m,n=1}^{\infty} a_{mn}}$ is Absolutely Convergent if $\displaystyle{\sum_{m,n=1}^{\infty} \mid a_{mn} \mid}$ converges, and Conditionally Convergent if $\displaystyle{\sum_{m,n=1}^{\infty} a_{mn}}$ converges and $\displaystyle{\sum_{m,n=1}^{\infty} \mid a_{mn} \mid}$ diverges.
• As expected, we saw that if the double series $\displaystyle{\sum_{m,n=1}^{\infty} a_{mn}}$ converges absolutely, then $\displaystyle{\sum_{m,n=1}^{\infty} a_{mn}}$ converges.
• We then began to look at the product of series on The Product of Two Series of Real Numbers page. We said that if $\displaystyle{\sum_{n=0}^{\infty} a_n}$ and $\displaystyle{\sum_{n=0}^{\infty} b_n}$ are two series of real numbers then the Product of these series is given by the partial sum sequence $(S_n)_{n=1}^{\infty}$ where for each $n \in \mathbb{N}$ we define:
(2)
\begin{align} \quad S_n = \left ( \sum_{i=0}^{n} a_i \right ) \left ( \sum_{j=0}^{n} b_k \right ) \end{align}
• To visualize how we determine the product of two series, consider the following array form of the terms in the expansion of the product:
(3)
\begin{align} \quad \begin{matrix} a_0b_0 & a_0b_1 & a_0b_2 & \cdots & a_0b_n & \cdots \\ a_1b_0 & a_1b_1 & a_1b_2 & \cdots & a_1b_n & \cdots \\ a_2b_0 & a_2b_1 & a_2b_2 & \cdots & a_2b_n & \cdots \\ \vdots & \vdots & \vdots & \ddots & \vdots & \ddots \\ a_nb_0 & a_nb_1 & a_nb_2 & \cdots & a_nb_n & \cdots \\ \vdots & \vdots & \vdots & \ddots & \vdots & \ddots \end{matrix} \end{align}
• Then each $S_n$ for $n \in \{0, 1, 2, ... \}$ is the sum of the top left $n+1$ by $n+1$ array of terms.
• We saw that if $\displaystyle{\sum_{n=0}^{\infty} a_n}$ and $\displaystyle{\sum_{n=0}^{\infty} b_n}$ both converge to $A$ and $B$ then the product of these series will also converge - specifically to the product $AB$.
• On The Cauchy Product of Two Series of Real Numbers page we looked at a different type of product between two series of real numbers. If $\displaystyle{\sum_{n=0}^{\infty} a_n}$ and $\displaystyle{\sum_{n=0}^{\infty} b_n}$ are two series of real numbers, then the Cauchy Product of these series is the following sum:
(4)
\begin{align} \quad \sum_{n=0}^{\infty} c_n = \sum_{n=0}^{\infty} \underbrace{ \left ( \sum_{k=0}^{n} a_kb_{n-k} \right )}_{c_n} \end{align}
• We then looked into the convergence of Cauchy products of series on the Convergence of Cauchy Products of Two Series of Real Numbers. We saw that if $\displaystyle{\sum_{n=0}^{\infty} a_n}$ and $\displaystyle{\sum_{n=0}^{\infty} b_n}$ both converge absolutely and converge to $A$ and $B$ respectively, then the Cauchy product $\displaystyle{\sum_{n=0}^{\infty} c_n}$ converges to $AB$.
• We also noted that if $\displaystyle{\sum_{n=0}^{\infty} a_n}$ converges absolutely and $\displaystyle{\sum_{n=0}^{\infty} b_n}$ only converges conditionally, say to $A$ and $B$ respectively, then $\displaystyle{\sum_{n=0}^{\infty} c_n}$ converges (not necessarily absolutely) to $AB$.
• Lastly, on The Cauchy Product of Power Series we looked at a nice result on Cauchy products for power series. We saw that if $\displaystyle{\sum_{n=0}^{\infty} a_nx^n}$ and $\displaystyle{\sum_{n=0}^{\infty} b_nx^n}$ are two power series that are absolutely converge to $A(x)$ and $B(x)$ respectively, then the Cauchy product $\displaystyle{\sum_{n=0}^{\infty} c_nx^n}$ converges to $A(x)B(x)$.