## Riemann Series Theorem

For the discussion of math. Duh.

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xepher
Posts: 83
Joined: Tue Mar 09, 2010 1:42 am UTC

### Riemann Series Theorem

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
Posts: 2370
Joined: Thu Nov 01, 2007 4:06 am UTC

### Re: Riemann Series Theorem

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.

Qaanol
The Cheshirest Catamount
Posts: 3069
Joined: Sat May 09, 2009 11:55 pm UTC

### Re: Riemann Series Theorem

++\$_ 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.
wee free kings