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.
diverges but 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
(1) converges and
(2) then also the series
(3) converges and has the same sum as (1).
Proof. Let be the sum of the (1). Then for each positive integer , there exists an integer such that . The partial sum of (3) may be written
When , we have
by the convergence of (1) to , and
by the condition (2). Therefore the whole partial sum will tend to , 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 |
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Canonical name | AddingAndRemovingParenthesesInSeries |
Date of creation | 2013-03-22 18:54:09 |
Last modified on | 2013-03-22 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 |