We are in the sequences section of our Freshman calculus class. One of the homework problems was to find whether the sequence converged or diverged. This sequence diverges, but it isn’t easy for a freshman to see.
I’ll discuss this problem and how one might go about explaining it to a motivated student. To make things a bit simpler, I’ll discuss the sequence instead.
Of course is periodic with a fundamental region so we will work with that region. Now we notice the following:
is a group with the usual operation of addition.
By , I mean the set where ; one can think of the analogue of modular arithmetic, or one might see the elements of the group .
Of course, to get additive inverses, we need to include the negative integers, but ultimately that won’t matter. Example: are just equal to themselves , etc. So, I’ll denote the representative of by .
Now if then ; for if then there would be integers so that which would imply that is a multiple of . Therefore there are an infinite number of in which means that the set has a limit point in the compact set which means that given any positive integer there is some interval of width that contains two distinct (say, greater than .)
This means that so there is some integers so that , etc. Therefore there is some multiple of in every interval of width . But was an arbitrary positive integer; this means that the are dense in . It follows that is dense in and hence cannot converge as a sequence.
Frankly, I think that this is a bit tough for most Freshman calculus classes (outside of, say those at MIT, Harvard, Cal Tech, etc.).