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Can someone help me verify if this is true?

When using the Alternating Series Test (AST), do I need to look at the absolute values of the terms and see if they converge to confirm that the series is absolutely convergent? If they don't converge, would the series be conditionally convergent?

Also, to prove if the absolute value of the series is convergent, what tests do I need to use? I've used the ratio and geometric tests, but are there others that are easier?

Thanks!!

Jee
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2 Answers2

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The alternating series test is only useful for telling when something conditionally converges. The series of absolute values is never alternating, so the AST can never be used to tell if the convergence is absolute.

Yes, your assessment is correct. If you can show convergence by the AST, but its series of absolute values does not converge, then the convergence is conditional. The classic example is the alternating harmonic series. On the other hand, if you demonstrated absolute convergence, then convergence is implied, so there's no need for the alternating series test.

I wouldn't say there are any that are "easier"; whether something is easy or useful often depends on constant. The ratio test is usually pretty easy when it works, as is the root test. In practice, you will probably use comparison to a known convergent or divergent series most often. Typical convergent series for comparison are geometric series and power series like $1/n^2.$ And the most useful divergent one is the harmonic series, though $\frac{1}{n\log n}$ can be useful when that fails.

spaceisdarkgreen
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  • Thanks for the great answer! On a side note, could I prove absolute convergence with the ratio test because the ratio test is taking the absolute value of a sub (n+1) / a sub n? – Jee May 02 '18 at 00:49
  • @Jee Yes, when the ratio test shows convergence, it shows absolute convergence. Conditionally convergent sequences will come up inconclusive (try it). Inconclusive could mean absolutely convergent, conditionally convergent or divergent, though. You are right that you can tell this since the ratio test only refers to the absolute value of coefficients. – spaceisdarkgreen May 02 '18 at 00:55
  • Okay thanks so much for the help! – Jee May 02 '18 at 01:03
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The definition of an absolutely convergent series is that the sum of the absolute values of the terms converges. You don't need to use that with the Alternating Series Test to show the sum is convergent, but you do if you want to show the sum is absolutely convergent. Another useful test for convergence is the comparison test. If you can find a convergent series that is larger term by term than the one you are testing, the one you are testing is convergent as well.

Ross Millikan
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