- #1
Lebombo
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Homework Statement
This is what I understand about Alternating Series right now:
If I have an alternate series, I can apply the alternative series test.
[itex]\sum(-1)^{n}a_{n}[/itex]
Condition 1: Nth term test on [itex]a_{n}[/itex]
Condition 2: 0 < [itex] a_{n+1} ≤ a_{n}[/itex]If condition 1 is positive or ∞, convergence is inconclusive, try another test.
If condition 1 is 0, then go to condition 2.
If condition 2 is false, then convergence is inconclusive, try another test.
If condition 2 is true, then series is convergent, go to absolute convergence test.Absolute convergence test
[itex]\sum |a_{n}|[/itex]
Choose a test to find convergence or divergence.
If convergent, Alternating series is absolutely convergent.
If divergent, Alternating series is conditionally convergent.So my questions come from applying Alternating Series Conditions 1 and 2.
Question 1:
If condition 1 is positive or ∞ (instead of 0), then the test is said to be inconclusive, but then what test can then be applied? It looks like Condition 1 is sort of the nth term test, so I would be inclined to think that a positive or ∞ would imply divergence. So if it doesn't imply divergence (only inconclusiveness), what then?
Question 2:
If condition 1 turns out to be 0 and condition 2 is tested and turns out false, the test is said to be inconclusive, but then, again, what test can be applied thereafter?