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By a multiple sequence of order $n\in\mathbb{N}^*$ I mean a function $a:~\mathbb{N}^n\longrightarrow\mathbb{K}~(=\mathbb{R}\text{ or }\mathbb{C})$ such that $$a(i_1,\cdots,i_n)=a_{i_1i_2\cdots a_n}$$ and its associated series $$\sum\limits_{i_n=0}^{\infty}\cdots\sum\limits_{i_1=0}^{\infty}a_{i_1i_2\cdots a_n}$$ Even less are the sequences: $\mathbb{Z}^n\longrightarrow\mathbb{K}$; even sequences like $a:~\mathbb{Z}\longrightarrow\mathbb{K}$ are very hard to find in books.

And any suggestions for books that studies such objects (multi-sequences, multi-series and series on $\mathbb{Z}$).

Michael Hardy
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user5402
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    Do you have a question? – Chris Eagle Jul 09 '13 at 20:22
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    I think multi-indices, and collections of numbers indexed by them, are used quite commonly... for example, Taylor's theorem in several variables (and more generally, occasions where parital derivatives come up) – Zev Chonoles Jul 09 '13 at 20:25
  • You can usually find examples of double series in basic analysis texts. The difficulties that arise in not always being able to interchange the order of summation make studying even higher dimensional sums undesirable, while generalizations of Cauchy's double series theorem are relatively simple and not very enlightening (at least in my opinion). But you still see plenty of double sequences and series. – grantfgates Jul 09 '13 at 20:25
  • @metacompactness: While I think you're incorrect about these particular objects being ignored, you might consider my answer on this question as to the phenomenon of "ignored" structures in mathematics in general. – Zev Chonoles Jul 09 '13 at 20:31
  • @ChrisEagle I'm looking for books that study such things and wondering why the notion of integral had been studied in higher dimensions whereas the notion of series hadn't. – user5402 Jul 09 '13 at 20:37
  • @ZevChonoles I'm not talking about any object that uses multi-indices like tensors; I'm talking about a complete theory of such sequences and series (convergence and limits, mutli-series of functions,...) – user5402 Jul 09 '13 at 20:41
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    You should look into the concept of nets. These are generalized sequences, which your multi-index material would fall into the category of. Nets are used mostly by analysts to study general topologies while keeping their intuition about sequences. For instance you could define the sum of a series of multi-indices via a net. – Joel Jul 09 '13 at 21:02
  • @Joel Great suggestion but what about multi-series? I want to do some "calculus" on these series like multiple power series, differentiation... does nets help in such situations or is there some others structures I should look at? – user5402 Jul 09 '13 at 21:31
  • The idea of a net will give you some way of interpreting convergence of one of these series. For instance instead of partial sums what you would have is a collection of subsets $F_1 \subset F_2 \subset \cdots$ for which $\cup F_n = Z^k$ and the collection of sums ${ \sum_{v \in F_n} a_{v} }$. If the limit of that new sequence of sums does not depend on the choice of subsets, then you can say that the sum converges. – Joel Jul 09 '13 at 21:38
  • Multiple power series have been considered by a lot of people. I suggest you look at the Wiki pages for Taylor series in several variables. – Joel Jul 09 '13 at 21:39
  • @Joel I suggest you convert your comments into an answer. – user5402 Jul 09 '13 at 21:47
  • Why the downvote?! Not every question should be of the form "prove that...." – user5402 Jul 09 '13 at 21:49
  • Hah ok. I guess I'm not very points centered. – Joel Jul 09 '13 at 22:02
  • One might prefer to get a title in the lines of "Are multiples sequences and series ignored?". The answer is readily seen to be "No, they aren't." – Pedro Jul 10 '13 at 01:43

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You should look into the concept of nets. These are generalized sequences, which your multi-index material would fall into the category of. Nets are used mostly by analysts to study general topologies while keeping their intuition about sequences. For instance you could define the sum of a series of multi-indices via a net.

The idea of a net will give you some way of interpreting convergence of one of these series. For instance instead of partial sums what you would have is a collection of subsets $F_1 \subset F_2 \subset \cdots$ for which $\cup F_n = \mathbb{Z}^k$ and the collection of sums $\{\sum_{v \in F_n} a_v\}$. If the limit of that new sequence of sums does not depend on the choice of subsets, then you can say that the sum converges.

Also multiple power series have been considered by a lot of people. I suggest you look at the Wiki pages for Taylor series in several variables.

Joel
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