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Mathematics
Calculus
Alternating Series, Testing for Convergence
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[QUOTE="axmls, post: 5442015, member: 488930"] That's odd. My textbook also says for all ##n##, however I checked Paul's notes online and he specifically points out that it only needs to be eventually decreasing, and your sequence does eventually strictly decrease. I would go with Paul's notes. After all, suppose that the sequence ##a_n## is not decreasing for ##1, 2, ... N## but that it is decreasing for all ##n > N## (and further suppose that ##\lim_{n \to \infty} a_n = 0##). Then certainly we could simply write the sum as $$\sum _{n=1} ^\infty (-1)^n a_n = \sum_{n=1} ^N (-1)^n a_n + \sum_{n = N+1} ^\infty (-1)^n a_n$$ Then certainly the first term is finite, and the second term converges by the alternating series test. Of course, you'd have to show that your sequence is in fact strictly decreasing after some ##N##, but intuitively that's certainly the case for ##\sin(x)## as ##x \to 0##. In this case, I'd say your function is strictly decreasing for ##n \geq 12##. I'd love to hear someone else's opinion, though. It's quite possible that the textbook intends to say this: get the series in a form such that it is always decreasing, even if you have to split it up into some finite sum and an infinite sum. [/QUOTE]
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Alternating Series, Testing for Convergence
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