I know this question has come up numerous times in the 10+ years I've been reading the listserv, but I was unable to find an example in a quick search of the archives.
Here is my question: Which series can students state as convergent or divergent without further proof?
It's been my understanding that the harmonic and alternating harmonic series are series that students can just state as "divergent" and "convergent," respectively, but are there others? For example, can they simply state (without further verification) that the latter converges conditionally, i.e., without actually mentioning that sum(1/n) diverges?
Also, could they give a reason such as, "sum((-1)^n / sqrt(n)) converges conditionally because it is an alternating p-series (0<p<=1)," or do they HAVE to show that it meets the conditions of the AST?
In the past, I've always just encouraged them to use the AST, so it's never really been an issue, and I've never really paid attention to the answer when others have asked this same question on the listserv. But now, for whatever reason, I'd like to know.
Speaking of the AST, do they actually have to PROVE the a_n are decreasing, or just STATE that they are? And do they have to provide a calculation that confirms that a_n -> 0 as n -> inf?