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. 
 A: What you are conjecturing is not correct in general. But it may be true for any specific $a_k$ and $b_k$. We need to know the specific terms are defined to prove anything about the specific case.
You want to look up the Riemann Rearrangement Theorem. In fact, the theorem states almost the exact opposite. From the Wikipedia article:

In mathematics, the Riemann series theorem (also called the Riemann rearrangement theorem) says that if an infinite series of real numbers is conditionally convergent, then its terms can be arranged in a permutation so that the new series converges to an arbitrary real number, or diverges.

It guarantees the existence of two permutations such that the series are not equal. Examples and proof is given in the Wikipedia article.
A: Well it's not true. In fact a conditionally cgt series can be rearranged to give any limit. Proof: take as many positive terms until you get over the desired limit, then take negative ones until you get below, etc etc.
For absolute convergence it is true. Both these are standard results.
