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We just learned about conditionally convergent series vs absolutely convergent series in my hs calculus class and I'm really confused about why conditionally convergent series are seemingly allowed to ignore the Commutative Property.

We were given this example in class.

$$1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}... = \ln(2)$$

We weren't shown why this is $\ln(2)$ but just that it was given.

Then he rearranged into

$$(1-\frac{1}{2})-\frac{1}{4}+(\frac{1}{3}-\frac{1}{6})-\frac{1}{8}+(\frac{1}{5}-\frac{1}{10})...$$

which equals

$$\frac{1}{2}-\frac{1}{4}+\frac{1}{6}-\frac{1}{8}+\frac{1}{10}... $$

he then factors out 1/2 which gives

$$\frac{1}{2}(1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}...)$$

Which includes the original series so the final conclusion is that the new series is

$$\frac{\ln(2)}{2}$$

Now here is my question : WHY???
This makes sense (sort of) but I learned in second grade that 1+2=2+1 , which is the Commutative Property, I don't understand why we can ignore that just because it's an infinite series. Are there any more clear examples or better ways of thinking about it so that it actually makes sense, or is it just one of those things that you have to take as given. Why is it considered "Conditionally Convergent" instead of "Divergent" ?

Thank you for the help

Prem
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Meezird
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    First, the word is commutative, not communitive. Second, there is no contradiction since the commutative property refers very specifically only to finitely many additions taking place. It says nothing about infinitely many additions taking place. As for proof that infinitely many may not be rearranged without potentially affecting the final outcome, your example is a standard one and the overall result is stated in the Riemann series theorem. – JMoravitz Feb 28 '23 at 17:49
  • "just because it's an infinite series." - Welcome to infinity! There are many properties that are true for finite sets, on which we build our intuitions, that fail to be true for infinite sets. As another instance, "a set can be put in one-to-one correspondence with a proper subset of itself" is utterly false for finite sets, but true for infinite ones. You're just going to have to develop a more accurate intuition for infinity, step-by-step. – JonathanZ Feb 28 '23 at 18:03

2 Answers2

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You can see this rigorously with the Riemann' theorem which states that any convergent series which doesn't converge absolutely can be rearranged to converge to any real number. The main issue with those series is that infinite series are at the end of the day nothing but limits of sequences and limits of sequences measure the value of the sequence for very large n. By permuting the terms you can "borrow" the terms that come later on to make the series smaller in total. The value of your series is the limit of partial summs which under permutation can be made smaller/larger in total by using the "later" terms to alter smaller terms. I believe the proof of the Riemann's theorem offers some insight but other than that it is quite paradoxical and unintuitive to pretty much everyone.

Boxonix
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The commutative property holds for TWO summands. By induction you can make it work for $n,$ but induction only works for integer number of arguments, and infinity is not an integer.

Igor Rivin
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