Two approaches to the product of two sums Let $s_n=\sum_{k=1}^n a_k$ and $t_n=\sum_{k=1}^n b_k$ be convergent real sequences/sums with limits $s=\sum_{k=1}^\infty a_k$ and $t=\sum_{k=1}^\infty b_k$. Since the limit of the product of two convergent real sequences is exactly the product of the respective limits, we know:
$$ (\sum_{k=1}^\infty a_k) \cdot (\sum_{k=1}^\infty b_k)=\lim_{n\to\infty} s_n \cdot t_n = \lim_{n\to\infty} \sum_{k,l=1}^n a_k \cdot b_l  $$
If we have respective absolute convergent sums $s_n$ and $t_n$, we know the Cauchy Product:
$$ \lim_{n\to\infty} \sum_{l=1}^n \sum_{k=1}^l a_k \cdot b_{l+1-k} =(\sum_{k=1}^\infty a_k) \cdot (\sum_{k=1}^\infty b_k) $$
In this case, it seems like:
$$ \lim_{n\to\infty} \sum_{l=1}^n \sum_{k=1}^l a_k \cdot b_{l+1-k} = \lim_{n\to\infty} \sum_{k,l=1}^n a_k \cdot b_l $$
My question: Is this true? And if so, why is the first intuitive approach to the product of sums not seen in standard textbooks?
Edit: I know that the Cauchy product has some more properties and more meaning, but that's not what I'm talking about. I want to talk about the product of sums with a new perspective, namely from a standpoint concerned about sequences.
 A: This is wrong:
$$\lim_{n\to\infty} \sum_{k=1}^n a_k \cdot b_{n+1-k} =(\sum_{k=1}^\infty a_k) \cdot (\sum_{k=1}^\infty b_k)$$
The correct expression is:
$$(\sum_{k=1}^\infty a_k) \cdot (\sum_{k=1}^\infty b_k) := \lim_{n\to\infty} \sum_{k=1}^n a_k b_{n+1-k}$$
Why? Because this is defining the Cauchy Product, which is being denoted by "$\cdot$".
In your version, you used "$\cdot$" in the same equation with two different meanings. In "$a_k \cdot b_{n+1-k}$", it means ordinary multiplication of the two real or complex values. But in $(\sum_{k=1}^\infty a_k) \cdot (\sum_{k=1}^\infty b_k)$ is denoting the Cauchy Product, which is a convolution of two series - a different operation entirely.
And that is the answer to your question. These are diffent operations. Your third equation is false. In your first equation, you are just talking about the multiplication of two numbers, and how multiplication interacts with limits. This is discussed in textbooks, which is why you were able to reproduce it here. But because it was so intuitive, you didn't really notice it.
But the Cauchy Product is something different. It is not just multiplying two series together. It is a type of summation that one encounters somewhat regularly, so it gets a special name and some study of its properties.
You will see many other cases of operations denoted by the same symbol. There are just too many possible operations out there to give unique symbolry to each of them. You wouldn't be able to remember them all. You determine which operation is meant by context.
