In class I was demonstrating the various open intervals of absolute convergence and gave the usual as an example of a series that converges at only. I mentioned that “ doesn’t even pass the divergence test”, which, as it turns out, is true. But why? (yes, it is easier to just use the ratio test and be done with it)
Well, I should have noted: if , then for some integer m, then for we have and one can see that this is a finite number times a number which is growing without bound. Hence the sequence of terms of the series grows without bound for any positive value of .