Cauchy product of several series

In summary, the conversation discusses the Cauchy product of several series and the difficulties in understanding the notation used for it in a Wikipedia article. The speaker explains how the notation is necessary for dealing with products of finite arbitrary many factors and how it can be complicated when dealing with multiple series. The conversation also touches on the challenges of implementing the notation in practice and the objective of finding a more efficient way to handle large integer exponents of Taylor series.
  • #1
m4r35n357
658
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TL;DR Summary
Explanation of notation or alternative source desired.
I am trying to make sense of the wikipedia article section regarding Cauchy product of several series. but am stuck right at the start because the notation used there is unfamiliar to me and not explained previously in the article.
The commas in ##\Sigma a_1, k_1## etc. mean nothing to me. Am I missing something obvious? Is there a better article anywhere (I haven't found one)?
 
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  • #2
If you want to consider products of finite arbitrary many factors, then you have to deal with it. There is no way to write it easier. E.g. if we had four series, we could write ##\sum_{k}a_k\, , \,\sum_{l}b_l\, , \,\sum_{m}c_m\, , \,\sum_{n}d_n##, but we have ##n## of them and no ##n## letters in the alphabet. In addition, it makes the formulas more complicated, as we would have terms ##a_{k}b_{l}c_{m}d_{n}## with ##(k,l,m,n) ## from a subset of indices of ##\mathbb{N}^4##. Hence we need to write ##a=a_1\, , \,b=a_2\, , \,c=a_3\, , \,d=a_4\, , \,\ldots ## as the different series elements. Now every one of them has an index, as it is a series. Here we have the similar problem: ##k,l,m,n## but with ##n## many series we run out of letters. Therefore we write ##k=k_1\, , \,l=k_2\, , \,m=k_3\, , \,n=k_4\, , \,\ldots## instead. Finally the comma in ##a_{j,k_j}## is only meant to distinguish the two indices: ##j-##th series with index ##k_j = 1,2,\ldots##
 
  • #3
Hmm, not sure I understand. Are you saying that the term(s) within the product/sum ##\Pi \Sigma## on the third line do not represent a formula that I can implement myself, because that is my objective (I just don't know what the term means).
 
  • #4
m4r35n357 said:
Hmm, not sure I understand.
Dito.
Are you saying that the term(s) within the product/sum ##\Pi \Sigma## on the third line ...
The third line is "and the ##n##th one converges. Then the series". I assume you mean the sixth.
... do not represent a formula that I can implement myself, ...
What? I have only explained the meaning of notation, and nothing about implementations.
... because that is my objective (I just don't know what the term means).
This is what I have explained, or at least tried to. ##\prod_{j=1^n}\left( \sum_{k_j=0}^\infty a_{j,k_j}\right)## is simply the product of ##n## series. The left hand side is a bit complicated, and cannot be implemented, since your algorithm won't halt. It is ##n## sums of products of ##n## factors, where every factor is from a different series, and the sum is over all possible combinations. It is the distributive law for ##n## factors and ##n## series.
 
  • #5
Heh, I meant the third line of equations, so you got the right part! So if I get you the term just represents all the combinations requred to form the RHS. In that case I would agree it gets too big to handle (numerically) for large numbers of series.

I am actually investigating whether there is a sensible way of doing (large) integer exponents of taylor series. This literal approach is the baseline, but maybe there is a better way. Real exponents are surprisingly simple, but are restricted to a powers of a positive base.

Thanks, I need to go away and think a bit more.
 

FAQ: Cauchy product of several series

1. What is the Cauchy product of several series?

The Cauchy product is a mathematical operation that combines two or more infinite series into a single series. It is also known as the Cauchy multiplication or Cauchy convolution.

2. How is the Cauchy product calculated?

The Cauchy product is calculated by multiplying the first term of the first series with all the terms of the second series, then multiplying the second term of the first series with all the terms of the second series, and so on. The resulting terms are then added together to form the new series.

3. What are the conditions for the Cauchy product to converge?

In order for the Cauchy product to converge, at least one of the series must converge absolutely. Additionally, the product of the absolute values of the coefficients of the two series must also converge.

4. Can the Cauchy product of two convergent series diverge?

Yes, it is possible for the Cauchy product of two convergent series to diverge. This can happen if the product of the absolute values of the coefficients of the two series does not converge.

5. What are some applications of the Cauchy product?

The Cauchy product is used in many areas of mathematics, including complex analysis, number theory, and probability theory. It is also used in physics and engineering to solve problems involving infinite series, such as in Fourier series and Laplace transforms.

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