Why does the harmonic series diverge but the p-harmonic series converge I am struggling understanding intuitively why the harmonic series diverges but the p-harmonic series converges. I know there are methods and applications to prove convergence, but I am only having trouble understanding intuitively why it is. I know I must never trust my intuition, but this is hard for me to grasp. In both cases, the terms of the series are getting smaller, hence are approaching zero, but they both result in different answers.
$$\sum_{n=1}^{\infty}\frac{1}{n}=1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\cdots = \text{diverges}$$   $$\displaystyle \sum_{n=1}^{\infty}\frac{1}{n^2}=1+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+\cdots =\text{converges}$$

If you convert the sum to an integral,
$$
\int_1^\infty \frac{1}{x^2} dx = -\frac{1}{x}|_1^\infty = -\frac{1}{\infty} + 1
$$
converges, but
$$
\int_1^\infty \frac{1}{x} dx = \ln x|_1^\infty = \ln \infty
$$
doesn't.
