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If we can pick any conditionally convergent series, $\sum_{n=1}^\infty a_n$, and pick any real number, $\ell$, and show that

$$\sum_{n=1}^\infty a_n = \ell,$$

then why is that not considered a contradiction, since we can show that this sum takes on any value, and any real number can be equal to any other real number? In my limited experience in math, usually something like this is considered a contradiction, and I had expected that such a sum would just be said to diverge or DNE.

It's really confusing me today.

Aaron Kirk
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    You are mis-quoting Riemann's Rearrangement Theorem. It should be $\sum_{n = 1}^\infty a_{\sigma(n)} = \ell$ for some permutation $\sigma$. – balddraz Feb 05 '21 at 04:30

1 Answers1

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The theorem does not say that $$\sum_{n=1}^\infty a_n = \ell,$$it says that if $$\sum_{n=1}^\infty a_n$$is conditionally convergent, for every $\ell \in \Bbb R$ there is a bijection $\sigma:\Bbb N \to \Bbb N$ such that $$\sum_{n=1}^\infty a_{\sigma(n)}=\ell.$$It's a different series. For $\sigma = {\rm Id}_{\Bbb N}$ you have the original value of the series. For other values, non-trivial permutations.

Ivo Terek
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