# Commutative law in conditionally convergent series.

"The problem of series that are not absolutely convergent is that you can't make arbitrary rearrangement of the terms."

However, my understanding was that due to the commutative law, we could write the following (c.f. Concrete Mathematics):

$$\sum_{k\in\mathbb{K}}{a_{k}}=\sum_{p(k)\in\mathbb{K}}{a_{p(k)}},$$

For any permutation function $p(k):\mathbb{Z}\to\mathbb{Z}, \space\forall k \in\mathbb{Z}$, in which all integers are mapped to an integer without duplicity, and for all sequences $a_{k}$.

Is this true even for conditionally converging series, if so, is my confusion based on the author of that answer's definition of "arbitrary rearrangement" (I'm assuming that a permutation is a form of "arbitrary rearrangement")?

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@Cocopuffs Thanks for the link! – Shaktal Jun 17 '12 at 14:47

The distributive, associative, and commutative laws on p. 30 of Concrete Mathematics, formulas (2.15)-(2.17), are stated only for finite $K$. You cannot freely extend them to infinite index sets.
So they are, my mistake. Is this because there is no function which maps the entirety of $\mathbb{N}$ to a rearrangement of $\mathbb{N}$ including every element once and only once? – Shaktal Jun 17 '12 at 14:43