The criteria for testing for convergence with the alternating series test, according to my book, is: Σ(-1)n-1bn With bn>0, bn+1 ≤ bn for all n, and lim n→∞bn = 0. My question is about the criteria. I'm running into several homework problem where bn is not always greater than bn+1, such as the following: Σ(-1)n sin(6π/n). This sequence is also not always greater than zero either (n=4 and n=5 make this negative), nor is it (-1)n-1 like the criteria says, but the series converges anyways. From n=6 to n=12, it appears that bn < bn+1. But my criteria says bn+1 ≤ bn for all n. Should this read as n→∞ instead of for all n? Am I missing something? What's with these apparent inconsistencies?