Given a (convergent or divergent) series $\sum a_n$, let's mark the sequence of points $(n,a_n)\in\mathbb R^2$Īnd join the consecutive points by straight segments. I like the following geometric interpretation. Exercises 11(b) and 12(b) may serve as illustrations.Įxercise 11(b) states that if $\sum_n a_n$ is a divergent series of positive reals, then $\sum_n a_n/s_n$ also diverges, where $s_n = \sum_=\infty.$$ The point we wish to make is this: No matter how we make this notion precise, the conjecture is false.
This notion of “boundary” is of course quite vague. One might thus be led to conjecture that there is a limiting situation of some sort, a “boundary” with all convergent series on one side, all divergent series on the other side-at least as far as series with monotonic coefficients are concerned. In Rudin's Principles of Mathematical Analysis, following Theorem 3.29, he writes: The following is a FAQ that I sometimes get asked, and it occurred to me that I do not have an answer that I am completely satisfied with.
