adding and removing parentheses in series

We consider series with real or complex terms.

  • If one groups the terms of a convergent seriesMathworldPlanetmathPlanetmath by adding parentheses but not changing the order of the terms, the series remains convergentMathworldPlanetmath and its sum the same. (See theorem 3 of the entry.)

  • A divergent series can become convergent if one adds an infinite amount of parentheses; e.g.
    1-1+1-1+1-1+- diverges but (1-1)+(1-1)+(1-1)+ converges.

  • A convergent series can become divergent if one removes an infinite amount of parentheses; cf. the preceding example.

  • 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.

  • If the series

    (a1++ar)+(ar+1++a2r)+(a2r+1++a3r)+ (1)

    converges and

    limnan= 0, (2)

    then also the series

    a1+a2+a3 (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  kr<n(k+1)r.  The partial sum of (3) may be written


    When  n, we have


    by the convergence of (1) to S, and


    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 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