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I was asked by an online portal to determine the convergence of this series: $$ \sum_{n=1}^\infty(-1)^nn^{-1}\ln(n+6).$$ Using the alternating series test, both conditions were fulfilled ($\lim_{n\to\infty} a_n=0$, and $|a_{n+1}|<|a_n|$).

The homework portal says that the result is convergent, but is not absolutely convergent. On most other problems of this type, the series I got asked seemed to be absolutely convergent when just these two conditions were fulfilled. What am I missing here?

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  • $\begingroup$ Wellcome to here $\endgroup$ – HK Lee Apr 8 '15 at 6:33
  • $\begingroup$ "On most other problems of this type, the series I got asked seemed to be absolutely convergent" Are you sure? You don't need the alternating series test if the series is absolutely convergent. I also must say your title is strange to me. What is inconclusive about the AST? $\endgroup$ – zhw. Apr 8 '15 at 6:35
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In problems like these we have the following theorem:

Theorem Suppose that we have a series $\sum a_n$ and either $a_n=(-1)^nb_n$ or $a_n=(-1)^{n+1}b_n$ or where $b_n\geq 0$ for all n. Then if,

  • $\lim_{n\to \infty } b_n=0$ and
  • $b_n$ is a decreasing sequence

the series $\sum a_n$ is convergent.

Let's apply this test for your series:

  • $a_n=(-1)^n\frac{\ln(n+6)}{n}$ and $b_n=\frac{\ln(n+6)}{n}$
  • $\lim_{n\to \infty } b_n=0$ because $n$ always dominates $\ln(n+a)$
  • $b_n$ is decreasing positive

As a consequence $\sum (-1)^n\frac{\ln(n+6)}{n}$ is convergent, and the series $\sum |a_n|=\sum b_n$is not absolutely convergent, and in the general case almost all alternating series are not absolutely convergent.

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  • $\begingroup$ Thanks for your response! Those were the tests I applied as well, but I did not know that alternating series are not absolutely convergent (indicated otherwise by the online question.) $\endgroup$ – Ferreroire Apr 10 '15 at 5:24
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The alternating series test does not give absolute convergence. The standard example is the alternating harmonic series $1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \frac{1}{5}- \frac{1}{6} +\ldots$, which is convergent by the alternating series test, but is not absolutely convergent.

Your example has $\sum |a_n| = \ln(7) + \frac{\ln(8)}{2} + \frac{\ln(9)}{3} + \frac{\ln(10)}{4} + \frac{\ln(11)}{5}+ \ldots $, which diverges by comparison with the harmonic series.

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