The Harmonic Series and p-Series

A few series come up so often that you should know at once whether they converge. Geometric series are one family. The harmonic series and the p-series are two more.

Its terms approach 0, so the nth term test says nothing. The function $f(x) = \frac{1}{x}$ is positive, continuous and decreasing for $x \ge 1$, so the integral test applies.

The integral diverges, so the harmonic series diverges. Its partial sums grow very slowly. The rectangles of height $\frac{1}{k}$ on $[k, k + 1]$ lie above $y = \frac{1}{x}$, so $S_n > \int_1^{n+1} \frac{dx}{x} = \ln(n + 1)$.

The partial sum first passes 5 at $n = 83$, and it first passes 10 at $n = 12{,}367$. No finite list of partial sums can show that a series diverges, so the test is needed.

The harmonic series is the case $p = 1$ of a whole family. The integral test decides every member at once.

For $p > 0$, $f(x) = x^{-p}$ is positive, continuous and decreasing on $[1, \infty)$. With $p \ne 1$,

When $p > 1$, $b^{1-p} \to 0$ and the integral is $\frac{1}{p - 1}$. When $p < 1$, $b^{1-p} \to \infty$ and the integral diverges. By the integral test, the p-series converges for $p > 1$ and diverges for $0 < p < 1$. The case $p = 1$ is the harmonic series. For $p \le 0$ the terms don't approach 0, so the series diverges by the nth term test.

In a p-series the variable is the base and $p$ is the exponent. In a geometric series such as $\sum \frac{1}{2^n}$ it's the other way around, and the two tests aren't interchangeable.

Giving the harmonic series alternating signs changes its behavior.

Its partial sums swing up and down, and the swings shrink.

They close in on $\ln 2 \approx 0.6931$. Unlike the harmonic series, the alternating harmonic series converges, and a test for alternating series proves it.