From A Calculus Problem to Generating Functions

On a warm late November morning, following a slow moving coal truck on Route 286, I said to myself: great, I will be late. I was suppose to observe Dan Burkett's class and it seemed like I could not full fill my obligation...

I was about ten minutes late. It was fortunate that the class was a review session therefore missing the first few minutes might not have been as catastrophic as compared to...

This talk, however, is not about my guilty feelings of being late nor about the evaluation. It is about, well, a student's question asked in that class. The question was: How to find the radius of convergence of

formula

Dan answered the question beautifully and the students were enlightened. Looking at the problem, Dan then talked to himself: does the series converge when |z| = 1/4 ? Paused for a moment, he said: I need to sit down to think about that.

After the observation, I had about one hour to kill. So I sat down and thought about Dan's question. This talk is on my thoughts in that hour, or more precisely, my current recollection of my thoughts in that hour, and some related events thereafter.