Cauchy product of several series

[m4r35n357]

TL;DR

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)?

 

[fresh_42]

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 $k_j = 1,2,\ldots$

 

[m4r35n357]

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).

 

[fresh_42]

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.

 

[m4r35n357]

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.