If $\sum{r_j}$ is an absolutely convergent series bounded by R, and the sequence $\{s_{j,k}\}$ is bounded by S and $\lim_{k \to \infty} s_{j,k} = 0$, then how do we show that $\lim_{n \to \infty} \sum^{\infty}_{j=0}r_j s_{j,k}$ go to $0$? By the answer here Under what condition we can interchange order of a limit and a summation? I can see that we can interchange the limit and the sum, but I was wondering what you need to be able to show this a bit more intuitively.
2026-03-28 10:17:36.1774693056
multiplying a convergent series by a bounded sequence
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The Dominated Convergence Theorem will justify this rigorously. If $|s_{j,k}|\le S$ always, then each series is dominated by $\sum S|r_j|$ which converges by hypothesis; therefore the limit can be taken inside the infinite series.