The criteria for testing for convergence with the alternating series test, according to my book, is:(adsbygoogle = window.adsbygoogle || []).push({});

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 readasn→∞instead offor all n?

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

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# I Alternating Series, Testing for Convergence

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