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[parent] sum function of series (Definition)

Let the terms of a series be real functions $ f_n$ defined in a certain subset $ A_0$ of $ \mathbb{R}$; we can speak of a function series. All points $ x$ where the series

$\displaystyle f_1+f_2+\cdots$ (1)

converges form a subset $ A$ of $ A_0$, and we have the sum function $ S\!:x\mapsto S(x)$ of (1) defined in $ A$.

If the sequence $ S_1,\,S_2,\,\ldots$ of the partial sums $ S_n = f_1\!+\!f_2\!+\cdots+\!f_n$ of the series (1) converges uniformly in the interval $ [a,\,b] \subseteq{A}$ to a function $ S\!:x\mapsto S(x)$, we say that the series converges uniformly in this interval. We may also set the direct

Definition. The function series (1), which converges in every point of the interval $ [a,\,b]$ having sum function $ S:x\mapsto S(x)$, converges uniformly in the interval $ [a,\,b]$, if for every positive number $ \varepsilon$ there is an integer $ n_\varepsilon$ such that each value of $ x$ in the interval $ [a,\,b]$ satisfies the inequality

$\displaystyle \vert S_n(x)-S(x)\vert < \varepsilon$
when $ n \geqq n_\varepsilon$.

The notion of uniform convergence of series can be extended to the series with complex function terms (the interval is replaced with some subset of $ \mathbb{C}$). The significance of the uniform convergence is therein that the sum function of a series with this property and with continuous term-functions is continuous and may be integrated termwise.



"sum function of series" is owned by pahio.
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See Also: uniform convergence of integral, sum of series

Also defines:  function series, sum function, uniform convergence of series

This object's parent.

Attachments:
Weierstrass' criterion of uniform convergence (Theorem) by pahio
termwise differentiation (Theorem) by Mathprof
theorems on complex function series (Theorem) by pahio
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Cross-references: continuous, property, complex function, inequality, integer, positive, function, interval, partial sums, sequence, converges, points, subset, real functions, series, terms
There are 10 references to this entry.

This is version 14 of sum function of series, born on 2004-09-24, modified 2006-09-24.
Object id is 6223, canonical name is SumFunctionOfSeries.
Accessed 9120 times total.

Classification:
AMS MSC: 40A30 (Sequences, series, summability :: Convergence and divergence of infinite limiting processes :: Convergence and divergence of series and sequences of functions)
 26A15 (Real functions :: Functions of one variable :: Continuity and related questions )
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