Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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":

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

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 $S=0$.

share|improve this question

1 Answer 1

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).

http://en.wikipedia.org/wiki/Riemann_series_theorem

share|improve this answer
    
@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
1  
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: groups.google.com/group/sci.math/msg/ae96aa939cad546a –  Dave L. Renfro Aug 1 '11 at 20:37

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.