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Can someone please tell me where this line of reasoning goes wrong?
False proof for the convergence of the alternating harmonic series:

Break the series $S = 1 - 1/2 + 1/3 - 1/4 + \dots$ into the following "subseries":

$S_1=1 - 1/2 - 1/4 - 1/8 - \dots$
$S_3=1/3 - 1/6 - 1/12 - 1/24 - \dots$
$S_5=1/5 - 1/10 - 1/20 - 1/40 - \dots$
$S_7=1/7 - 1/14 - 1/28 - 1/56 - \dots$

First, it seems that no term from the original series occurs in more than one subseries. If a term did occur in more than one, then we would have $z \cdot 2^i= y \cdot 2^j$, where $z$ and $y$ are odd. So $z \cdot 2^{i-j}=y$, and the only way this can happen is if $i=j$ and $z=y$.

Second, it seems that every term from the original series can be found in one of the subseries. Take any integer $k$ and decompose it into $k=(2^i) \cdot z$, where $z$ is odd. Then the $k$th term of the original series can be found in the $i+1$ term of $S_z$.

Therefore $S=S_1 + S_3 + S_5 + S_7 + \dots$, and all the $S_i=0$, so $S=0$.

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Since the series is not absolutely convergent, rearrangements of the terms can affect the sum. What you have is a very pretty rearrangement of the sum that evaluates to $0$. Other rearrangements can be found to have the sum be any number we like, including $\pm \infty$ (see the link below).

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@lhf: Thanks for providing the reference. – Shaun Ault Aug 1 '11 at 18:25
Wow... That's an amazing theorem. So commutativity is yet another thing that can break down when you introduce infinity? – zodiak770 Aug 1 '11 at 18:36
The Wikipedia references don't seem to be very well selected. I recommend the following, neither of which (at present) is mentioned in the Wikipedia article: Kerry Smith McNeill, "Rearrangement of series", Pi Mu Epsilon Journal 10 (1997), 547-555. Peter Rosenthal, "The remarkable theorem of Levy and Steinitz", American Mathematical Monthly 94 #4 (April 1987), 342-351. Also, see the following post for some surprising Baire category results for "almost all rearrangements" of a conditionally convergent series: – Dave L. Renfro Aug 1 '11 at 20:37

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