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Σ(-1)

^{n-1}b

_{n}

With b

_{n}>0, b

_{n+1}≤ b

_{n}for all n, and lim n→∞b

_{n}= 0.

My question is about the criteria. I'm running into several homework problem where b

_{n}is not always greater than b

_{n+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 b

_{n}< b

_{n+1}. But my criteria says b

_{n+1}≤ b

_{n}

*for all n*.

Should this read

*as*

*n→∞*instead of

*for all n*?

Am I missing something? What's with these apparent inconsistencies?