Talk:Cauchy product

Convergence and Mertens' theorem

I changed the counterexample $\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{n}}$, which can't start at $n=0$ and whose Cauchy square does converge, though conditionally, contrary to the stated. I put the counterexample $\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{\sqrt {n+1}}}$ fixing but issues. It is well defined and its Cauchy square $\sum b_{n}$ does diverge. Indeed, as stated, $b_{n}$ doesn't converge to zero. 85.54.190.251 (talk) 12:38, 22 January 2011 (UTC)

I also changed Spanish entry and noticed that the French and German entries have essentially the same counterexample, with brief proofs of the divergence. 85.54.190.251 (talk) 12:46, 22 January 2011 (UTC)

Cesaro's theorem

I changed the formulation of Cesaro's theorem. I believe the theorem was intended to be read in a way that would make it corect but there was a notational conflict with the rest of the article rendering it liable to be misunderstood. MathHisSci (talk) 12:15, 15 March 2010 (UTC)

Notation and references

I've removed the "unreferenced" tag and added a reference to Apostol. This includes most of the content, but not Cesaro's theorem; a further reference would be good. I've tagged the article "attention needed", as I find the notation C(x,y) somewhat confusing, and the use of $\sum x$ strikes me as too informal. I'm not sure whether it would be better to give a clear definition of "C(x,y)", or rewrite the article to avoid using that notation. Jowa fan (talk) 02:51, 28 April 2011 (UTC)

Update: I've cleaned up the notation somewhat; it would be nice if someone can proofread this and make sure I haven't introduced any silly errors. There seems to be some confusion as to whether the "Cauchy product" refers to the sequence or the series or both. It bothers me that the lead paragraph and the section "series" seem to say much the same; can anyone think of a nice way of rewriting the lead without using mathematical notation? The sections "Generalizations" and "Relation to convolution of functions" seem to be unsourced. Jowa fan (talk) 02:16, 30 April 2011 (UTC)

Hello Jowa fan, I think i will proofread the article today evening; my talk page indicates that it would be a good idea to finally do something useful here anyways. Good day, --Mathmensch (talk) 17:51, 4 March 2014 (UTC)

Summations

I added this piece:

________________________________________________________________________________________________

If the product comes from two finite summations, up to n, the Cauchy Product is defined as

$\left(\sum _{k=0}^{n}a_{k}\right)\cdot \left(\sum _{k=0}^{n}b_{k}\right)=\sum _{k=0}^{2n}\sum _{i=0}^{k}a_{i}b_{k-i}-\sum _{k=0}^{n-1}(a_{k}\sum _{i=n+1}^{2n-k}b_{i}+b_{k}\sum _{i=n+1}^{2n-k}a_{i})$

if (and only if) a_k and b_k are defined with k from 0 to 2n.

It is quite interesting the fact that if n goes to infinity this summation product become the series Cauchy product. ________________________________________________________________________________________________

and a user reverted the page because, he said:

23:41, 21 November 2011‎ Jowa fan (talk | contribs)‎ (8,534 bytes) (Undid revision 461802692 by 46.17.97.92 (talk) remove new section: no sources, and it doesn't make sense) (undo)

so I replied

09:47, 22 November 2011‎ 46.17.97.92 (talk)‎ (9,053 bytes) (Undid revision 461850889 by Jowa fan (talk) This formula comes from the lessons of calculus from my university (moscow state university) in elec.engin. Why it is a nonsense?) (undo)

then he reverted again the page, writing:

13:24, 22 November 2011‎ Jowa fan (talk | contribs)‎ (8,534 bytes) (Undid revision 461914058 by 46.17.97.92 (talk) 1. It's not clear what c_k equals; 2. See policy at WP:SOURCE) (undo)

I want to say first of all I read the WP:SOURCE.

The formula comes from lesson notes I took at university.

In these days I was bustling about with this formulas so I arrived here.

The beautifulness of a mathematical formula is that anyone can verify it.

I took a piece of paper and I tried to puzzle over the formula. I made the case n=3.

On the first member we have:

$\left(\sum _{k=0}^{3}a_{k}\right)\cdot \left(\sum _{k=0}^{3}b_{k}\right)$

or in other words:

$(a_{0}+a_{1}+a_{2}+a_{3})(b_{0}+b_{1}+b_{2}+b_{3})$

on the second member we have 2 pieces:

$\sum _{k=0}^{6}\sum _{i=0}^{k}a_{i}b_{k-i}$

that generate this:

$k=0\Rightarrow a_{0}b_{0}$

$k=1\Rightarrow a_{0}b_{1}+a_{1}b_{0}$

$k=2\Rightarrow a_{0}b_{2}+a_{1}b_{1}+a_{2}b_{0}$

$k=3\Rightarrow a_{0}b_{3}+a_{1}b_{2}+a_{2}b_{1}+a_{3}b_{0}$

$k=4\Rightarrow a_{0}b_{4}+a_{1}b_{3}+a_{2}b_{2}+a_{3}b_{1}+a_{4}b_{0}$

$k=5\Rightarrow a_{0}b_{5}+a_{1}b_{4}+a_{2}b_{3}+a_{3}b_{2}+a_{4}b_{1}+a_{5}b_{0}$

$k=6\Rightarrow a_{0}b_{6}+a_{1}b_{5}+a_{2}b_{4}+a_{3}b_{3}+a_{4}b_{2}+a_{5}b_{1}+a_{6}b_{0}$

and the second piece:

$\sum _{k=0}^{2}(a_{k}\sum _{i=4}^{6-k}b_{i}+b_{k}\sum _{i=4}^{6-k}a_{i})$

that generate this:

$k=0\Rightarrow a_{0}b_{4}+a_{0}b_{5}+a_{0}b_{6}+b_{0}a_{4}+b_{0}a_{5}+b_{0}a_{6}$

$k=1\Rightarrow a_{1}b_{4}+a_{1}b_{5}+b_{1}a_{4}+b_{1}a_{5}$

$k=2\Rightarrow a_{2}b_{4}+b_{2}a_{4}$

Because of the minus in front of the second piece, this delete the superabundant terms of the first piece.

It is interesting that if we increase n by 1 all the formula "shifts" by one row and it is not necessary to delete the two "side" terms on k=4. We need to delete from k=5.

If n is increased by 2 we'll start to delete from k=6 and so on.

If n→∞ we don't need to cancel anything and we have the Cauchy product.

In my opinion this could be "wikipediafied". What do you think?

46.17.97.92 (talk) 14:38, 22 November 2011 (UTC)

I agree that the formula is correct: the left hand side equals the right hand side. There is no problem there. The issue is that I don't think it belongs on this page: I don't believe that it's actually called "Cauchy product". The point of the Cauchy product for infinite series is that when you sum an infinite series, the answer may depend on the order of summation. So the Cauchy product defines $c_{n}$ to be something, then you sum in the order $c_{1}+c_{2}+\cdots$. With finite sums, the answer is the same no matter what the order, so there's no need for a Cauchy product. In your formula, there isn't a definition of "$c_{n}$", and there isn't a clear way to define it, because the choice of $a_{k}$ and $b_{k}$ is arbitrary for $k>n$.

The reason why I gave a link to WP:SOURCE is that I want to verify whether anyone uses the name Cauchy product for your formula. I agree that the formula is correct; the part that doesn't make sense to me is whether it belongs on this page.

I'll ask at Wikipedia talk:WikiProject Mathematics what other people think about this. Jowa fan (talk) 23:14, 22 November 2011 (UTC)

I don't think this formula is relevant to this article. It should probably be in some other article on finite series. My objection is that this formula isn't a Cauchy product; as far as I'm concerned, Cauchy products are only defined for infinite series.

In addition, I believe that some of the text is wrong. The article claims that it's interesting that as n tends to infinity, the right-hand side converges to the Cauchy product. As formal power series, convergence in the adic topology is trivial. Convergence in the usual topology (as a series of real numbers, not as a formal power series) is not always true. Indeed, the article already gives the example of $a_{n}=b_{n}=(-1)^{n+1}/{\sqrt {n+1}}$.

I'm editing the article to remove these statements. Can anyone think of a better home for the remaining formula? Ozob (talk) 19:45, 27 November 2011 (UTC)

Transferred this formula to the article Summation#Identities. J.P. Martin-Flatin (talk) 18:22, 12 November 2015 (UTC)

Semigroup Algebras

User:TakuyaMurata recently reinstated a paragraph on semigroup algebras that I had deleted. In my view, this concept is too advanced for the main audience of this article: first-year B.S. students in mathematics or second-year B.S. students in engineering. The relation to semigroup algebras would make sense in article Convolution (e.g., in section "Discrete convolution"), which is more advanced, but it does not belong in an entry-level article such as this one. What does the community think about it? J.P. Martin-Flatin (talk) 11:52, 14 November 2015 (UTC)

Who decided the scope of the article? I understand the engineering students may not be interested in semigroup algebras; those readers should just skip the section. -- Taku (talk) 12:58, 14 November 2015 (UTC)

Cesaro's theorem citation

Cesaro's theorem is currently unsourced. It traces back to Cesaro. See Silverman's book "On the Definition of the Sum of a Divergent Series", p.27, Thm. J for a modern English statement. He cites "Cesaro: Bull des Sciences math, t. XIV 1890". I do not presently have an inclination to figure out Wikipedia's citation system enough to edit the article with this information, but I wanted to at least put it somewhere public. 2603:8000:A303:F8EC:1845:84BC:A5A5:6EBE (talk) 07:56, 3 May 2022 (UTC)