Riemann Series Theorem

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xepher
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Joined: Tue Mar 09, 2010 1:42 am UTC

Riemann Series Theorem

Postby xepher » Wed Mar 02, 2011 6:42 pm UTC

So I've been looking at the Riemann Series Theorem recently, where when the terms of a conditionally convergent sequence are rearranged, they can form a different sum than the original arrangement.

Where's the proof that shows this holds for all conditionally convergent sequences? Does it?

++$_
Mo' Money
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Joined: Thu Nov 01, 2007 4:06 am UTC

Re: Riemann Series Theorem

Postby ++$_ » Wed Mar 02, 2011 7:14 pm UTC

Wikipedia actually has a proof.

The proof basically just amounts to "Pick positive terms from the series (starting with the largest in absolute value) until you have a partial sum greater than the desired limit; then pick negative terms until you have a partial sum less than the desired limit; then pick positive terms again until you have a partial sum greater than the desired limit; etc."

There are a few details that need to be checked -- in particular, that you always are able to "get to the other side" of the limit using only a finite number of terms, and that the series that results actually converges to the desired limit. They're not bad.

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Qaanol
The Cheshirest Catamount
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Re: Riemann Series Theorem

Postby Qaanol » Wed Mar 02, 2011 7:39 pm UTC

++$_ wrote:Pick positive terms from the series (starting with the largest in absolute value)

You can leave the positive terms in the order you found them, and the negative terms in the order you found them, and the construction still works.
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