All were quick to solve, until one simple question arose, concerning a trivial problem in Calculus 1 (or Analysis 1, as we name it here). The problem is:
out of the cold I just can't find counterexamples unless I've already thought about that problem before (as a generic property, some times I can do it).
This is something I was already aware, but this morning was a big hit. I kept working during the day on my usual tasks, but on my way to take the train back home I had a few minutes to spare a thought or two while going there. But just took me 1 minute, from my office to the university side door.
Let , then , which satisfies the conditions of the statement, and does not converge.
Anyway, I was not happy with this solution, as it was over our level in our first course. A more basic example not involving series should be easy to get. In the afternoon I had asked a friend the question through GTalk, but he was away, and as we picked it up again he wanted to know if I had some solution to it. I told him that indeed, but I was not happy. He then asked if the partial sums of the harmonic series were not enough (this is another name for my sequence ). As I told him that they were not, he suggested using logarithms. Aha!
Let , then by applying the mean value theorem,
where is a value between and . And , which implies that this sequence also satisfies the conditions and clearly diverges.
In conclusion, I was able to find the counter-examples, but it took me far more time than needed. Is speed that important?
While thinking about this problem, I remembered a book recommendation I was given a long time ago: Counterexamples in Analysis (buy on Amazon link). I never took the chance to read it... Maybe the time has arrived!
9 programming books I have read and somewhat liked...
And e Appears from Nowhere
C code juicer: detecting copied programming assignments
Cron, diff & wget: Watch changes in a webpage
8 reasons for re-inventing the wheel as a programmer
Approximating images with randomly placed translucent triangles