Alternating series convergence test - doesn't seem conclusive? 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?
 A: 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.
A: 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.
