by Shakey » Fri Jul 23, 2010 7:51 pm
This is actually something I tried working on at some point, but it presents many difficulties. Currently SpaceTime cannot input a series like Series(1/(1+e^n),n,1,Infinity) and return convergent or divergent. Here's what one would need to do:
1. Test for the limit of the terms going to 0, i.e. Limit(1/(1+e^n)) = 0? If the limit is not 0, then the answer is clearly divergent.
2. Assuming the limit is 0, this is where the hard part begins. One now needs to parse the terms to figure out exactly what is going on. Perhaps if one can determine a particular form of the expression one can apply the root test for example, but this then becomes a difficult task in parsing (although the function Part() can greatly help in this regard).
3. Failing all standard tests (usually coming from a College Calculus 2 course), the real fun begins with applying various inequalities in order to compare the series to one that we know either diverges or converges. This I believe is more of an art than a science, since I haven't really seen any software out there that can do this kind of thing.
Related question: Testing for whether a function is in L^1, i.e. integrable, is another related question that would involve comparing the function to another.
Stephen
