A question related to the Alternating Harmonic Series

Question

It's a well known result that the Alternating Harmonic Series : $$1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + ... = \ln(2)$$ But, this is because of "Conditional Convergence". I saw somewhere that if you re-arrange the terms in the series like this : $$1 + \frac{1}{3} - \frac{1}{2} + \frac{1}{5} + \frac{1}{7} - \frac{1}{4} +... = \frac{3}{2} \ln(2)$$ When I first saw it, my mind was blown. I couldn't accept it. How can the result change just by re-arranging the terms? My guess was that it is somewhat related to the $\mathbb{Z^+}$ paradox where the number of even integers is $\frac{1}{2}$ of all the numbers and $\frac{1}{3}$ when you arrange it in some other way(Not the theory but the logic behind it). Can anybody please explain why does the result of the Alternating Harmonic Series change when you re-arrange the terms. Any help is appreciated. Thanks in Advance