Q&A: Intuiting Edge Cases of Convergence and Divergence

Cross-posted from here.

Question

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?

Answer

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 $x \to \infty.$