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Infinite Series: rearrangement of Terms

  1. Dec 8, 2005 #1
    Hello,
    I could use a big helping hand in trying to understand an example from a text.
    Let's say I have a convergent series:

    [tex]S=\sum_{n=1}^{\infty}{a_n}[/tex]

    Okay, so now:

    [tex]\sum_{n=1}^{\infty}{a'_n}[/tex]

    is a rearrangement of the series where no term has been moved more than 2 places.

    So, the exercise is to show that the rearranged series has the same sum and is also convergent.

    A partial sum of the original series:

    [tex]S_N=a_1+a_2+\ldots +a_N = S_{N-2}+a_{N-1}+a_N[/tex]

    A partial sum of the rearranged series:

    [tex]S'_N=a'_1+a'_2+\ldots +a'_N = S'_{N-2}+a'_{N-1}+a'_N[/tex]

    I think I need to somehow bound
    [tex]S'_N[/tex]
    but I'm not sure how.

    Can someone steer me in the right direction?
    Thanks,
    Bailey
     
    Last edited: Dec 8, 2005
  2. jcsd
  3. Dec 8, 2005 #2

    shmoe

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    Try to bound [tex]|S'_N-S_N|[/tex]
     
  4. Dec 8, 2005 #3
    Okay, is this on the right track?

    So,

    [tex]S_n = a_1 + a_2 + \ldots + a_n[/tex]

    and

    [tex]S'_n[/tex] is a rearrangement of the series where the terms move no more than 2 places.

    Then

    [tex]|S'_n - S_n| \leq |a_{n-1}| + |a_n| + |a_{n_1}| + |a_{n_2}|[/tex]

    where [tex] n < n_1 < n_2[/tex]

    From that we get:

    [tex]S_n - (|a_{n-1}| + |a_n| + |a_{n_1}| + |a_{n_2}|) \leq S'_n \leq S_n + (|a_{n-1}| + |a_n| + |a_{n_1}| + |a_{n_2}|)[/tex]

    but since the original series converges: [tex]a_n \rightarrow 0[/tex]

    Therefore, [tex]S_n \leq S'_n \leq S_n[/tex] or [tex]S_n = S'_n[/tex]

    Is this correct?

    How do I show that the new series converges as well?
    Thanks,
    Bailey
     
  5. Dec 8, 2005 #4

    shmoe

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    What values of n do you think this equation is true?
     
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