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# series

Given a sequence of numbers (real or complex) $\{a_{n}\}$ we define a sequence of partial sums $\{S_{N}\}$, where $S_{N}=\sum_{{n=1}}^{N}a_{n}$. This sequence is called the series with terms $a_{n}$. We define the sum of the series $\sum_{{n=1}}^{\infty}a_{n}$ to be the limit of these partial sums. More precisely

$\sum_{{n=1}}^{\infty}a_{n}=\lim_{{N\to\infty}}S_{n}=\lim_{{N\to\infty}}\sum_{{% n=1}}^{N}a_{n}.$ |

In a context where this distinction does not matter much (this is usually the case) one identifies a series with its sum, if the latter exists.

Traditionally, as above, series are infinite sums of real numbers. However, the formal constraints on the terms $\{a_{n}\}$ are much less strict. We need only be able to add the terms and take the limit of partial sums. So in full generality the terms could be complex numbers or even elements of certain rings, fields, and vector spaces.

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40-00*no label found*

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## Comments

## summation over arbitrary index set?

This entry is too dry for me. Anybody willing to write something about the subject above? Cheers

## Re: summation over arbitrary index set?

For real numbers it only makes sense to consider countable index sets. If you take an uncountable set of positive real numbers and you want to define their "sum", it turns out that it is always infinite.. and the argument is that you can always find a countable subset whoose sum diverges (in the usual sense)..

There are many ways to see this, and one of them is:

Suppose you have an uncountable set of positive numbers X.

Consider the following subsets of R^+ (the set of positive real numbers): [1, \infty[; [1/2,1]; [1/3, 1]; [1/4, 1] ... [1/n, 1] ...

These subsets cover R^+, so at least one of them ([1/n, 1] for example) has an infinite number of elements of X. Now take a countable set of this infinite set and consider its series. All its elements are greater than 1/n, so the series diverges in the usual sense.

Yeah, an entry about this would be nice.. If I have time and nobody else does it I'll write it later today.

## Re: summation over arbitrary index set?

I may write one in the next couple days. I would define it as follows (and I'm basically just ripping off Rudin): if A is any set and f:A-[0,infty] is a function into the non-negative extended real numbers, then the "sum"

\sum_{a\in A}f(a)

is defined to be

\sup\{sum_{a\in F}f(a):F\subseteq A,F finite\}. If you do it this way, then it turns out that the above "sum" is equal to the integral of f over A with respect to the counting measure. Then you can (naturally?) generalize to complex-valued functions in l^1(A) by defining the unordered sum to be the Lebesgue integral of the function over A. I suppose this may be a bit contrived, but I kind of like it.

## Re: summation over arbitrary index set?

> For real numbers it only makes sense to consider

> countable index sets.

Except when all but a countable number of them are zero,

in which case we can define their sum to be the sum of

the countably many non-zero elements. This doesn't

contradict anything said above, just a way of generalizing

the definition.

The reason any of this is interesting is because we encounter

situations where such a generalized definition allowing

sums over uncountable sets is useful in certain situations.

For instance, in non-separable Hilbert (or, more generally,

Banach) spaces, we can have an uncountable number of basis

vectors and can express any vector in the space as a sum

over these basis elements if we define sums over uncountable

sets in the way described above.

> Yeah, an entry about this would be nice.. If I have time and

> nobody else does it I'll write it later today.

I already wrote an entry on this topic two years ago:

http://planetmath.org/?op=getobj;from=objects;id=7698

## Re: summation over arbitrary index set?

Yeah, that's it! :) Now it would be nice to also have the definition Keenan was referring to..

## Re: summation over arbitrary index set?

The definition of the sum as the supremum of sums over finite

subsets is to be found in the entry I wrote. The formulation as

Lesbegue integrals with a discrete measure is in the entry Boris

wrote:

http://planetmath.org/encyclopedia/SupportOfIntegrableFunctionWithRespec...

## Re: summation over arbitrary index set?

Thanks for pointing those entries out.

## Re: summation over arbitrary index set?

Thank you all for your replies. I think it'd be nice to see a link to the old (and beautiful) entry by rspuzio directly on the page about series, imho. Or maybe even merge it into it?