Convergence tests: Do/can they all have to hold? Anyone know if all the convergence tests are such that if one holds (e.g. claims that the series is convergent), then the others give the same outcome?
Not all convergence tests work for all problems though?
 A: There is a huge set $X$ of "all" sequences. This set is the disjoint union of the set $C$ of convergent sequences and the set $D$ of divergent sequences. Any convergence criterion ${\cal P}_i$ defines a certain subset $P_i\subset C$, and any divergence criterion ${\cal Q}_k$ defines a certain subset $Q_k\subset D$. The laws of logic guarantee that $P_i\cap Q_k=\emptyset$ for all $i$ and $k$; but this is all you can say in general.
Confronted with a sequence ${\bf x}:=(x_n)_{n\geq1}$ the best you can do is making a guess whether ${\bf x}\in C$, or ${\bf x}\in D$, and then checking whether ${\bf x}\in P_i$ (resp. ${\bf x}\in Q_k$) for one of the $P_i$, resp. $Q_k$, known to you.
A: Some convergence tests will only tell you one direction. The integral test, for example is approximating your series with either a lower bound or upper bound. If your series diverges, the lower bound may still converge. You have to look at the phrasing for each test you want to use and see if it's (implies) or (if and only if). 
Consider the root test, for example. If the root ratios are less than one, you guarantee convergence. But convergence only guarantees that he root ratios are less than or equal to one. 
Obviously you won't have two tests disagreeing. If test A says your series converges, test B will either agree or be inconclusive. 
