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 $A(x)B(x)$.