Q&A: Intuiting Edge Cases of Convergence and Divergence

by Justin Skycak on

Cross-posted from here.


It is known that $\displaystyle\sum_{n=1}^{\infty} \frac{1}{n^{1.000001}}$ converges while $\displaystyle\sum_{n\text{ is a prime number}}\frac{1}{n}$ diverges. Though we can logically prove these results, I found it difficult to explain their intuition to my students. How could it be done?


For me, the intuition just comes from the integral test (which is itself intuitive since a series is just a Riemann sum of rectangles with unit width).

  • The $n$th prime is asymptotically $n \ln n$ (see here for intuition on that), and the integral of $\dfrac{1}{x \ln x}$ is $\ln \ln x,$ whose end behavior is divergence as $x \to \infty.$
  • But as soon as that exponent in the denominator exceeds $1,$ the integral is a reciprocal power function, whose end behavior is convergence as $x \to \infty.$