The Product of Two Series of Real Numbers
Consider two series of real numbers, $\displaystyle{\sum_{n=0}^{\infty} a_n}$ and $\displaystyle{\sum_{n=0}^{\infty} b_n}$. Then we can consider the product of these two series:
(1)We can organize the terms of the expanding the product on the righthand side above in the following array:
(2)It is not clear how exactly we should sum up the terms in this array since the original multiplication involved multiplication of infinite sums. One such way is defined (generically) below.
Definition: Let $\displaystyle{\sum_{n=0}^{\infty} a_n}$ and $\displaystyle{\sum_{n=0}^{\infty} b_n}$ be two series of real numbers. The Product of these two series denoted $\displaystyle{\left ( \sum_{n=0}^{\infty} a_n \right ) \left ( \sum_{n=0}^{\infty} b_n \right )}$ is given by the partial sum sequence $\displaystyle{S_n = \sum_{i=0}^{n} \left ( \sum_{j=0}^{n} a_ib_j \right ) = \left ( \sum_{i=0}^{n} a_i \right ) \left ( \sum_{j=0}^{n} b_j \right )}$. |
This type of series multiplication tells us exactly how the add the terms in the array above. $S_0$ is given by summing up the red terms below:
(3)$S_1$ is given by summing up the orange terms below:
(4)$S_2$ is given by summing up the yellow terms below:
(5)In general, $S_n$ is given by summing up all terms in the $n + 1$ by $n+1$ top-left subarray above.
We will now look at a nice theorem which tells us if the two series in the product converge to $A$ and $B$, then the product of the two series will also converge to the product, $AB$.
Theorem 1: If $\displaystyle{\sum_{n=0}^{\infty} a_n}$ and $\displaystyle{\sum_{n=0}^{\infty} b_n}$ converge to $A$ and $B$ respectively, then the product $\displaystyle{\left ( \sum_{n=0}^{\infty} a_n \right ) \left ( \sum_{n=0}^{\infty} b_n \right )}$ converges to $AB$. |
- Proof: Let $(A_n)_{n=0}^{\infty}$ and $(B_n)_{n=0}^{\infty}$ be the sequences of partial sums for $\displaystyle{\sum_{n=0}^{\infty} a_n}$ and $\displaystyle{\sum_{n=0}^{\infty} b_n}$ respectively. If $S_n$ is the sequence of partial sums for the product of these two series, then:
- Since the sequences $A_n$ and $B_n$ converge to $A$ and $B$ as $n \to \infty$, we have that $S_n$ converges to $AB$ as $n \to \infty$. Therefore: