$A = \sum_{n=0}^\infty a_n$ and $b_n \to B$ implies $\sum_{k=0}^n a_k b_{n-k} \to AB$

This question is motivated by the answer With $y_n$ a sequence of real numbers, prove that if $y_n=x_{n-1}+2x_{n}$ converges then $x_n$ also converges, where essentially the following fact is used:

Let $A = \sum_{n=0}^\infty a_n$ be an absolutely convergent series, and $(b_n)$ a convergent sequence, $b_n \to B$. Then $$ \lim_{n \to \infty} \sum_{k=0}^n a_k b_{n-k} = A B \, . $$

This is not too difficult to prove (sketch): Write $$ \sum_{k=0}^n a_k b_{n-k} = B \sum_{k=0}^n a_k + \sum_{k=0}^n a_k \bigl( b_{n-k} - B \bigr) $$ The first sum converges to $AB$. For $\varepsilon > 0$, split the second sum into two parts $$ \sum_{k=0}^{n-N} a_k \bigl( b_{n-k} - B \bigr) + \sum_{k=n-N+1}^n a_k \bigl( b_{n-k} - B \bigr) \\ = \sum_{k=0}^{n-N} a_k \bigl( b_{n-k} - B \bigr) + \sum_{j=0}^{N-1} a_{n-j} \bigl( b_j - B \bigr) $$ where $N$ is chosen such that $\lvert b_n - B \rvert < \varepsilon$ for $n \ge N$. The first part can be estimated by $\varepsilon \sum_{n=0}^\infty |a_n|$, and the second (finite) sum converges to zero.

Now I am fairly sure that this is not new and must have been done before. However, I could not find a reference. So my question is: Is there a name for the above statement, or is there some "well-known theorem" for which this is just a special case?

The term $ \sum_{k=0}^n a_k b_{n-k}$ reminds me of the Cauchy product, but nothing is given about $\sum b_n$ here. Or is it perhaps related to summation methods for series?

(Or is it so trivial that everybody just knows it?)

In electronics, we call such a sum : the numerical convolution of two signals. And like you've remarked, it is a Cauchy product.

This theorem was proved by Franz Mertens. What's missing from the above proof, is supposing that at least one of the series has to be absolutely convergent. Otherwise, the estimation of the first part of the second sum, isn't justified. A counterexample is given in the Wikipedia link.

As it turns out, the statement is equivalent to Merten's theorem about the Cauchy product of two series (with the first series being absolutely convergent).

We define $$ \begin{align} \beta_0 &= b_0 \\ \beta_n &= b_n - b_{n-1} \text{ for } n \ge 1 \, . \end{align} $$ and $\gamma_n = \sum_{k=0}^n a_k \beta_{n-k}$.

Then $b_n = \beta_0 + \ldots + \beta_n$ and $B = \sum_{n=0}^\infty \beta_n$. A simple calculation shows that $$ \sum_{k=0}^n a_k b_{n-k} = \sum_{k=0}^n \gamma_k \, . $$

Therefore $\lim_{n \to \infty} \sum_{k=0}^n a_k b_{n-k} = A B $ is equivalent to $ \sum_{n=0}^\infty \gamma_n = A B$, and that is precisely he contents of Merten's theorem.