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We know the Cauchy product of two series $\big( \sum_n a_n \big) \big(\sum_n b_n \big) = \sum_j c_j$ where $c_j = \sum_{k=0}^j a_jb_{k-j}$. How do I write the Cauchy product of three series $\big( \sum_n a_n \big) \big(\sum_n b_n \big) \big(\sum_n c_n \big)$?

I tried $\big(\sum_n a_n \big)\Big[ \big(\sum_n b_n \big)\big(\sum_n c_n \big)\Big]$ and using the Cauchy product inside the big bracket and then distribute $a_n$ inside but then it became a mess. Any thoughts? Thanks in advance.

jimjim
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epsilon-delta
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  • "We know the product of two series..." , actually no, we don't know that at all, there are infinitely many way of defining products of series, see https://math.stackexchange.com/questions/1871203/how-to-multiply-two-infinite-series-correctly – jimjim Mar 10 '24 at 14:19

1 Answers1

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I think the most clean way is

$$ \big( \sum_n a_n \big) \big(\sum_n b_n \big)\big( \sum_n c_n \big) = \sum_k d_k $$ where $$ d_k=\sum_{i+j+l=k}a_ib_jc_l $$

Safwane
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Exodd
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