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I understand the basic idea of Conditionally Convergent (some infinitely long series can be made to converge to any value by reordering the series). I just do not understand how this could possibly be true. I think it defies common sense and seems like a clear violation of the Commutative Law.

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I believe, in particular, you are referring to the following theorem: en.wikipedia.org/wiki/Riemann_series_theorem –  Tyler Dec 16 '10 at 19:25
    
"it defies common sense" - You ought to read this book... as mentioned by the three excellent answers you've gotten, infinite sums don't have to behave the same way as finite sums. –  J. M. Dec 17 '10 at 1:22
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up vote 21 down vote accepted

What Riemann's Rearrangement Theorem on convergent series tells you is precisely that these "infinite sums" don't behave in the same way as the "usual" sums do.

The key is to understand that just because we can add two numbers together, and from that (through the use of the associative law) we can add any finite quantity of numbers together, we do not have a way to "add an infinite quantity of numbers" together.

So series are not really sums in the usual sense, they are limits, and they are limits of sequences. So commutativity and associativity of sums are not really at play, because series are not really sums in the usual sense.

Remember, $\sum_{i=1}^{\infty}a_n = L$ does not mean that you add infinitely many numbers and you get $L$, what it means is that the sequence $s_1,s_2,s_3,\ldots,s_n,\ldots$ converges to $L$, where $s_k = a_1+a_2+\cdots+a_k$ is the $k$th partial sum. So, for every $\epsilon\gt 0$ there exists $N\gt 0$ such that for all $n\geq N$, $|s_n-L|\lt \epsilon$. We are not adding up the infinitely many terms, we are saying that we can get arbitrarily close to $L$ if we add enough (but still only finitely many) of the terms.

Now, we can certainly play the usual games of commutativity and associativity so long as we only modify finitely many of the terms of the sequence $s_1,s_2,s_3,\ldots$, without modifying the limit: remember that the limit of the sequence only depends on what happens "eventually", not any changes you make to the beginning of the sequence. But that means that you need to keep the sequence intact from some point on. So you can only move around your original terms $a_n$ up to some point (up to some $n_0$) without having to worry about potentially changing the limit. But if you allow yourself to move the numbers around so much that you change the values of arbitrarily large partial sums, then you no longer have any reason to expect the new sequence to have the same limit as the old one (it could, but it doesn't have to, because you are now looking at an entirely different sequence). Which means that the usual ideas of "associativity" and "commutativity" of finite sums (you can add them up in any order and still get the same answer) no longer necessarily work for "infinite sums" (because they are not really sums, they are limits).

That is how Riemann's Rearrangement Theorem works: it modifies all the terms of the sequence so that you get a new limit.

(Moron has given you the standard example of why you cannot play associative and commutative games with infinite sums the way you do with finite sums, so I don't have to)

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When it comes to infinity, most common sense goes out the window. Cantor had a hard time convincing some of his peers...

For me what was surprising is that absolutely convergent series can be reordered in any fashion and still give the same result!

When begining to learn, one naturally considers the series $\displaystyle 1 - 1 + 1 - 1 \dots$

This can be written as

$\displaystyle 1 - (1-1) - (1-1) \dots = 1$

or

$\displaystyle (1-1) + (1-1) + \dots = 0$

Now it should not be a (big) surprise if we could reorder some series differently to give multiple possible values. From there it should not be a big surprise if there were some series which could give any possible value.

The proof of Riemann Rearrangement Theorem might help reduce the surprise :-)

Hope that helps.

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The commutative law holds for finitely many terms. There's no reason to expect it to hold for infinitely many terms unless the series converges absolutely. That's what absolute convergence is for. In mathematics, you can't assume that the infinite analogues of finite statements are always true; you have to prove it, because often they aren't.

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The other more detailed answers are good, but I think this gets to the heart of the question, of why it’s not “a clear violation of the commutative law”: because the commutative law doesn’t ever claim to cover infinite series. –  Peter LeFanu Lumsdaine Dec 16 '10 at 22:15
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It deserves to be better known that there are simple cases where one can give closed forms for some rearrangements of alternating series. Here are a couple of interesting examples based on results of Schlömilch in 1873. Many further results can be found in classical textbooks on infinite series, e.g. those by Bromwich and Knopp.

Let $\rm\ H^m_n\ $ be the rearrangement of the alternating harmonic series $\rm\ 1 - 1/2 + 1/3 - 1/4 +\: \cdots\ $ obtained by taking consecutive groups of $\rm\:m\:$ positive terms and $\rm\:n\:$ negative terms. Then

$$\rm H^m_n\ =\ log\ 2 + \frac{1}2\ \lim_{k\to\infty}\ \int^{\:mk}_{nk}\frac{1}x\ dx\ =\ \log 2 + \frac{1}2 \log\frac{m}n $$

Similarly rearranging Lebniz's sum $\rm\ L\ =\ \pi/4\ =\ 1 - 1/3 + 1/5 - 1/7 +\: \cdots\ $ yields

$$\rm L^m_n\ =\ \frac{\pi}4 + \frac{1}2\ \lim_{k\to\infty}\ \int^{\:mk}_{nk}\frac{1}{2x-1}\ dx\ =\ \frac{\pi}4 + \frac{1}4 \log\frac{m}n $$

Thus as $\rm\:m\:$ varies we obtain infinitely many rearrangements with distinct sums.

The proof of the general theorem underlying these results is quite simple - using nothing deeper than the integral test. See Beigel: Rearranging Terms in Alternating Series, Math. Mag. 1981.

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