Infinite Series: rearrangement of Terms

  • Thread starter MrBailey
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  • #1
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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:

Answers and Replies

  • #2
shmoe
Science Advisor
Homework Helper
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Try to bound [tex]|S'_N-S_N|[/tex]
 
  • #3
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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
 
  • #4
shmoe
Science Advisor
Homework Helper
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MrBailey said:
Therefore, [tex]S_n \leq S'_n \leq S_n[/tex] or [tex]S_n = S'_n[/tex]
What values of n do you think this equation is true?
 

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