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Suppose I have sums $A= \sum_i a_i$ and $B= \sum_i b_i$, and I define a product $A *B= \sum_i a_i b_i$, then are there results on:

  1. Relating convergence of $A$ and $B$ to that of $A*B$?
  2. Relating boundedness?
  3. Anything else, if somehow related to the above two concepts?
tryst with freedom
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  • Hmm . Close votes without any explanation. I don't understand what the problem with this question is.. – tryst with freedom May 07 '23 at 06:33
  • I believe you need absolute convergence for one of the sums in order to have the dot product converge. – Sean Roberson May 07 '23 at 06:34
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    Not my close vote, but this is an extremely unfocused question, and you've given zero evidence of having tried to figure out anything on your own. – JonathanZ May 07 '23 at 06:46
  • What would constitute as me trying to figure out something on my own here? I haven't seen any book discuss results related to this topic in my readings, so I don't even have a starting point. If you search up "product of series" on google, then most probably you would find something about cauchy product than this.@JonathanZsupportsMonicaC – tryst with freedom May 07 '23 at 06:51
  • https://math.stackexchange.com/questions/3041681/product-of-corresponding-terms-in-two-infinite-series – JonathanZ May 07 '23 at 07:01
  • https://math.stackexchange.com/questions/4060568/show-inner-product-of-two-infinite-sums-is-convergent-or-divergent – JonathanZ May 07 '23 at 07:02

1 Answers1

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I do not have the ability to comment, but the comment above is incorrect. As a counter-example, consider $A = B = \sum_n 1/n$ for which $A*B = \sum_n 1/n^2$ does converge. Neither sum absolutely converge, but their "product" does.

mildboson
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