real analysis - Consider two conditionally convergent series which are reorderings of each other, I conjecture them equal. Is there any tools I can use to prove this?

I have two conditionally convergent series $A = \sum^{\infty}_{k=1}a_{k}$ and $B = \sum^{\infty}_{k=1}b_{k}$ where $a_{k} = b_{P(k)}$ for a particular bijective map $P:\mathbb{N}\to\mathbb{N}$. I conjecture that they are equal, is there any useful results I can use to help me prove this.

Edit: I know the Riemann Rearrangement Theorem, but I am conjecturing that my series are in fact equal. I have two series, I can show they have the same terms and I have numerical evidence to suggest they are equal.