User:Mgkrupa/Generalizations of Series Proposal

The importance of Series in mathematics has led to many generalizations of notion. Generalizations include asymptotic series, sums assigned to divergent series, series with elements in topological groups (and topological vector spaces in particular), and series with uncountably many terms.

Asymptotic series
Asymptotic series, otherwise asymptotic expansions, are infinite series whose partial sums become good approximations in the limit of some point of the domain. In general they do not converge, but they are useful as sequences of approximations, each of which provides a value close to the desired answer for a finite number of terms. The difference is that an asymptotic series cannot be made to produce an answer as exact as desired, the way that convergent series can. In fact, after a certain number of terms, a typical asymptotic series reaches its best approximation; if more terms are included, most such series will produce worse answers.

Divergent series
Under many circumstances, it is desirable to assign a limit to a series which fails to converge in the usual sense. A summability method is such an assignment of a limit to a subset of the set of divergent series which properly extends the classical notion of convergence. Summability methods include Cesàro summation, (C,k) summation, Abel summation, and Borel summation, in increasing order of generality (and hence applicable to increasingly divergent series).

A variety of general results concerning possible summability methods are known. The Silverman–Toeplitz theorem characterizes matrix summability methods, which are methods for summing a divergent series by applying an infinite matrix to the vector of coefficients. The most general method for summing a divergent series is non-constructive, and concerns Banach limits.

Summations over arbitrary index sets
Definitions may be given for sums over an arbitrary index set $I$. There are two main differences with the usual notion of series: first, there is no specific order given on the set $I$; second, this set $I$ may be uncountable. The notion of convergence needs to be strengthened, because the concept of conditional convergence depends on the ordering of the index set.

If $$a:I \mapsto G$$ is a function from an index set $I$ to a set $G$, then the "series" associated to $$a$$ is the formal sum of the elements $$a(x) \in G $$ over the index elements $$x \in I$$ denoted by the
 * $$\sum_{x \in I} a(x).$$

When the index set is the natural numbers $$I=\mathbb{N}$$, the function $$a:\mathbb{N} \mapsto G$$ is a sequence denoted by $$a(n)=a_n$$. A series indexed on the natural numbers is an ordered formal sum and so we rewrite $$\sum_{n \in \mathbb{N}}$$ as $$\sum_{n=0}^{\infty}$$ in order to emphasize the ordering induced by the natural numbers. Thus, we obtain the common notation for a series indexed by the natural numbers
 * $$\sum_{n=0}^{\infty} a_n = a_0 + a_1 + a_2 + \cdots.$$

Families of non-negative numbers
When summing a family {ai}, i ∈ I, of non-negative numbers, one may define


 * $$\sum_{i\in I}a_i = \sup \Bigl\{ \sum_{i\in A}a_i\,\big| A \text{ finite, } A \subset I\Bigr\} \in [0, +\infty].$$

When the supremum is finite, the set of i ∈ I such that ai > 0 is countable. Indeed, for every n ≥ 1, the set $$ A_n = \{ i \in I \,:\, a_i > 1/n \}$$ is finite, because


 * $$ \frac 1 n \, \textrm{card}(A_n) \le \sum_{i\in A_n} a_i \le \sum_{i\in I}a_i < \infty.$$

If I&thinsp; is countably infinite and enumerated as I = {i0, i1,...} then the above defined sum satisfies


 * $$\sum_{i \in I} a_i = \sum_{k=0}^{+\infty} a_{i_k},$$

provided the value ∞ is allowed for the sum of the series.

Any sum over non-negative reals can be understood as the integral of a non-negative function with respect to the counting measure, which accounts for the many similarities between the two constructions.

Abelian topological groups
Let a : I → X, where I&thinsp; is any set and X&thinsp; is an abelian Hausdorff topological group. Let F&thinsp; be the collection of all finite subsets of I, with F viewed as a directed set, ordered under inclusion with union as join. Define the sum S&thinsp; of the family a as the limit


 * $$ S = \sum_{i\in I}a_i = \lim \Bigl\{\sum_{i\in A}a_i\,\big| A\in F\Bigr\}$$

if it exists and say that the family a is unconditionally summable. Saying that the sum S&thinsp; is the limit of finite partial sums means that for every neighborhood V&thinsp; of 0 in X, there is a finite subset A0 of I&thinsp; such that


 * $$S - \sum_{i \in A} a_i \in V, \quad A \supset A_0.$$

Because F&thinsp; is not totally ordered, this is not a limit of a sequence of partial sums, but rather of a net.

For every W, neighborhood of 0 in X, there is a smaller neighborhood V&thinsp; such that V − V ⊂ W. It follows that the finite partial sums of an unconditionally summable family ai, i ∈ I, form a Cauchy net, that is, for every W, neighborhood of 0 in X, there is a finite subset A0 of I&thinsp; such that


 * $$\sum_{i \in A_1} a_i - \sum_{i \in A_2} a_i \in W, \quad A_1, A_2 \supset A_0.$$

When X&thinsp; is complete, a family a is unconditionally summable in X&thinsp; if and only if the finite sums satisfy the latter Cauchy net condition. When X&thinsp; is complete and ai, i ∈ I, is unconditionally summable in X, then for every subset J ⊂ I, the corresponding subfamily aj, j ∈ J, is also unconditionally summable in X.

When the sum of a family of non-negative numbers, in the extended sense defined before, is finite, then it coincides with the sum in the topological group X = R.

If a family a in X&thinsp; is unconditionally summable, then for every W, neighborhood of 0 in X, there is a finite subset A0 of I&thinsp; such that ai ∈ W&thinsp; for every i not in A0. If X&thinsp; is first-countable, it follows that the set of i ∈ I&thinsp; such that ai ≠ 0 is countable. This need not be true in a general abelian topological group (see examples below).

Unconditionally convergent series
Suppose that I = N. If a family an, n ∈ N, is unconditionally summable in an abelian Hausdorff topological group X, then the series in the usual sense converges and has the same sum,


 * $$\sum_{n=0}^\infty a_n = \sum_{n \in \mathbf{N}} a_n.$$

By nature, the definition of unconditional summability is insensitive to the order of the summation. When ∑an is unconditionally summable, then the series remains convergent after any permutation σ of the set N of indices, with the same sum,


 * $$\sum_{n=0}^\infty a_{\sigma(n)} = \sum_{n=0}^\infty a_n.$$

Conversely, if every permutation of a series ∑an converges, then the series is unconditionally convergent. When X&thinsp; is complete, then unconditional convergence is also equivalent to the fact that all subseries are convergent; if X&thinsp; is a Banach space, this is equivalent to say that for every sequence of signs εn = ±1, the series


 * $$\sum_{n=0}^\infty \varepsilon_n a_n$$

converges in X.

Series in topological vector spaces
If X is a Topological Vector Space (TVS) and $$\left( x_{\alpha} \right)_{\alpha \in A}$$ is a (possibly uncountable) family in X then this family is summable if the limit $$\lim_{H \in \mathcal{F}(A)} x_{H}$$ of the net $$\left( x_H \right)_{H \in \mathcal{F}(A)}$$ converges in X, where $$\mathcal{F}(A)$$ is the directed set of all finite subsets of A directed by inclusion $$\subseteq$$ and $$x_H := \sum_{i \in H} x_i$$.

It is called absolutely summable if in addition, for every continuous seminorm p on X, the family $$\left( p \left( x_{\alpha} \right) \right)_{\alpha \in A}$$ is summable. If X is a normable space and if $$\left( x_{\alpha} \right)_{\alpha \in A}$$ is an absolutely summable family in X, then necessarily all but a countable collection of $$x_{\alpha}$$'s are 0. Hence, in normed spaces, it is usually only ever necessary to consider series with countably many terms.

Summable families play an important role in the theory of nuclear spaces.

Series in Banach and semi-normed spaces
The notion of series can be easily extended to the case of a seminormed space. If xn is a sequence of elements of a normed space X and if x is in X, then the series Σxn converges to x in  X if the sequence of partial sums of the series $$\left( \sum_{n=0}^N x_n \right)_{N=1}^{\infty}$$ converges to x in X; to wit,
 * $$\left\| x - \sum_{n=0}^N x_n \right\| \to 0$$

as N → ∞.

More generally, convergence of series can be defined in any abelian Hausdorff topological group. Specifically, in this case, Σxn converges to x if the sequence of partial sums converges to x.

If (X, |&middot;|)&thinsp; is a semi-normed space, then the notion of absolute convergence becomes: A series $$\sum_{i \in \mathbf{I}} x_i $$ of vectors in X&thinsp; converges absolutely if


 * $$ \sum_{i \in \mathbf{I}} \left| x_i \right| < +\infty$$

in which case all but at most countably many of the values $$\left| x_i \right|$$ are necessarily zero.

If a countable series of vectors in a Banach space converges absolutely then it converges unconditionally, but the converse only holds in finite-dimensional Banach spaces (theorem of ).

Well-ordered sums
Conditionally convergent series can be considered if I is a well-ordered set, for example, an ordinal number α0. One may define by transfinite recursion:


 * $$\sum_{\beta < \alpha + 1} a_\beta = a_{\alpha} + \sum_{\beta < \alpha} a_\beta$$

and for a limit ordinal α,


 * $$\sum_{\beta < \alpha} a_\beta = \lim_{\gamma\to\alpha} \sum_{\beta < \gamma} a_\beta$$

if this limit exists. If all limits exist up to α0, then the series converges.