$$\log2=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}-\frac{1}{6}+\cdots$$ $$\frac{\log2}{2}=\frac{1}{2}-\frac{1}{4}+\frac{1}{6}-\frac{1}{8}+\frac{1}{10}-\frac{1}{12}+\cdots$$ Adding these two convergent series gives $$\log2 + \frac{\log2}{2}=1+\frac{1}{3}+\frac{1}{5}+\frac{1}{7}\cdots+2\left(-\frac{1}{4}-\frac{1}{8}-\frac{1}{12}-\frac{1}{16}\cdots\right)=\log{2}$$
What did I do wrong?
The series involved are not absolutely convergent, so they cannot be rearranged with the same result; that is, there isn't an infinite associative property for conditionally convergent series.
In fact, it's generally true that a series is absolutely convergent if and only if all its rearrangments converge to the same limit. It's also true that, given a real number and a conditionally convergent series, there is a rearrangement of the series convering the to the number.