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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

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

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