Alternating Series Test No Divergence?

[jdawg]

Homework Statement

Hey! So I just have a quick question. In my notes I wrote down that the alternating series test only proves absolute or conditional convergence, but can not prove divergence. Is this true or did I misunderstand my professor?

Homework Equations

The Attempt at a Solution

 

[gopher_p]

Homework Statement

Hey! So I just have a quick question. In my notes I wrote down that the alternating series test only proves absolute or conditional convergence, but can not prove divergence. Is this true or did I misunderstand my professor?

Homework Equations

The Attempt at a Solution

The alternating series test says nothing about absolute convergence; both $\sum\frac{(-1)^n}{n}$ and $\sum\frac{(-1)^n}{n^2}$ converge by the alternating series test. The latter is absolutely convergent while the former is not.

Given an alternating series, i.e. a series of the form $\sum\limits_{n=0}^\infty(-1)^na_n$ with $a_n>0$, the alternating series test has you check two things; (1) that $\{a_n\}$ is a (eventually) decreasing sequence and (2) that $\lim\limits_{n\rightarrow\infty}a_n=0$. If either or both of these checks fails, then the alternating series test is technically inconclusive. There are some alternating series which fail the alternating series test and are convergent and some which fail and are divergent.

Put another way, the alternating series test says something along the lines of, "If we have an alternating series and (1) and (2) are true, then the series converges". It makes no claim if one or both of (1) and (2) are false, and it never concludes that a series is divergent.

But ...

In the event that a series fails part (2) of the test, then we can say that $\lim\limits_{n\rightarrow\infty}a_n\neq 0$. It's not hard to see that this implies that $\lim\limits_{n\rightarrow\infty}(-1)^na_n\neq 0$, and so the series is divergent by the $n$th-term test/test for divergence (or whatever name your text/lecturer gives to that test). So you can learn enough about a series in the process of conducting the alternating series test to conclude, as a result of a different test, that the series is divergent.

 

[LCKurtz]

The alternating series test is a test for convergence. But if the test fails to show convergence, that doesn't imply divergence. It might be convergent anyway and the alternating series test just isn't adequate to show it. All you can say is that the alternating series test failed to show convergence.