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Summation Convergency tests by 3-condition test

For each of the following series, tell whether or not you can apply the 3-condition test (i.e. the alternating series test). If you can apply this test, enter D if the series diverges, or C if the series converges. If you can't apply this test (even if you know how the series behaves by some other test), enter N.
1. Σ ((-1)^n)/(n^5)
2. Σ (((-1)^n)((n^3)+1))/((n^4)+1)
3. Σ (((-1)^n)(cos n))/(n^2)
4. Σ (((-1)^n)((n^4)+2n))/((n^3)-1)
5. Σ (((-1)^n)((n^3)+1))/((n^3)+7)
6. Σ (((-1)^n)((n^10)+1))/(e^n)

(all are from n=1 to infinity)

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1 Answer

The Alternating Series Test

∑(-1)nBn converges when the following two conditions are met:

(i) lim Bn = 0 and (ii) {Bn} is (eventually) decreasing.

Note: AST doesn't apply when either of the conditions is not met, and so never is a test for divergence.  If the first condition isn't met, then the "n-th term test" will show divergence.

Demonstrating convergence is all that can be accompished for the series that meet the conditions.  For the ones that don't meet the conditions you will have to enter 'N'.  The way the problem is stated I don't think 'D' will be a correct response for any of these.

Hope this helps.  If you're having trouble on a particular application of this test, please be specific with what you have tried, and I or another tutor will be happy to assist you.