adding and removing parentheses in series
We consider series with real or complex terms.

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If one groups the terms of a convergent series^{} by adding parentheses but not changing the order of the terms, the series remains convergent^{} and its sum the same. (See theorem 3 of the http://planetmath.org/node/6517parent entry.)

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A divergent series can become convergent if one adds an infinite amount of parentheses; e.g.
$11+11+11+\mathrm{\dots}$ diverges but $(11)+(11)+(11)+\mathrm{\dots}$ converges. 
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A convergent series can become divergent if one removes an infinite amount of parentheses; cf. the preceding example.

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If a series parentheses, they can be removed if the obtained series converges; in this case also the original series converges and both series have the same sum.

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If the series
$({a}_{1}+\mathrm{\dots}+{a}_{r})+({a}_{r+1}+\mathrm{\dots}+{a}_{2r})+({a}_{2r+1}+\mathrm{\dots}+{a}_{3r})+\mathrm{\dots}$ (1) converges and
$\underset{n\to \mathrm{\infty}}{lim}{a}_{n}=\mathrm{\hspace{0.33em}0},$ (2) then also the series
${a}_{1}+{a}_{2}+{a}_{3}\mathrm{\dots}$ (3) converges and has the same sum as (1).
Proof. Let $S$ be the sum of the (1). Then for each positive integer $n$, there exists an integer $k$ such that $$. The partial sum of (3) may be written
$${a}_{1}+\mathrm{\dots}+{a}_{n}=\underset{s}{\underset{\u23df}{({a}_{1}+\mathrm{\dots}+{a}_{kr})}}+\underset{{s}^{\prime}}{\underset{\u23df}{({a}_{kr+1}+\mathrm{\dots}+{a}_{n})}}.$$ When $n\to \mathrm{\infty}$, we have
$$s\to S$$ by the convergence of (1) to $S$, and
$${s}^{\prime}\to 0$$ by the condition (2). Therefore the whole partial sum will tend to $S$, Q.E.D.
Note. The parenthesis expressions in (1) need not be “equally long” — it suffices that their lengths are under an finite bound.
Title  adding and removing parentheses in series 

Canonical name  AddingAndRemovingParenthesesInSeries 
Date of creation  20130322 18:54:09 
Last modified on  20130322 18:54:09 
Owner  pahio (2872) 
Last modified by  pahio (2872) 
Numerical id  13 
Author  pahio (2872) 
Entry type  Topic 
Classification  msc 40A05 
Related topic  EmptySum 