The harmonic series is the sum of the reciprocals of all positive integers:

$$1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \frac{1}{6} + \cdots = \sum_{n=1}^{\infty} \frac{1}{n}$$

It diverges — the sum grows without bound, even though the terms shrink to zero.

Why it diverges

The classic proof groups terms in powers of 2:

$$1 + \frac{1}{2} + \underbrace{\frac{1}{3} + \frac{1}{4}}{\geq 2 \times \frac{1}{4} = \frac{1}{2}} + \underbrace{\frac{1}{5} + \frac{1}{6} + \frac{1}{7} + \frac{1}{8}}{\geq 4 \times \frac{1}{8} = \frac{1}{2}} + \underbrace{\frac{1}{9} + \cdots + \frac{1}{16}}_{\geq 8 \times \frac{1}{16} = \frac{1}{2}} + \cdots$$

Each group of $2^k$ terms sums to at least $1/2$. There are infinitely many such groups, so the sum exceeds any bound.

Nicole Oresme gave this proof around 1350 — one of the earliest proofs of divergence of any series.

How slowly it diverges

The harmonic series diverges, but barely. The partial sums grow logarithmically:

$$H_N = 1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{N} \approx \ln N + \gamma$$

where $\gamma \approx 0.5772$ is the Euler-Mascheroni constant. To reach a partial sum of 10, you need about 12,367 terms. To reach 20, you need about 272 million. To reach 100, you’d need roughly $10^{43}$ terms.

The name

“Harmonic” comes from music. The harmonics of a vibrating string have frequencies proportional to $1, 2, 3, 4, \ldots$ — the reciprocals $1, 1/2, 1/3, 1/4, \ldots$ are the corresponding wavelengths. The series sums these wavelengths.

What changes when you modify the exponent

The harmonic series is $\sum 1/n^1$. What about $\sum 1/n^s$ for other values of $s$?

The boundary is at $s = 1$: convergent for $s > 1$, divergent for $s \leq 1$.

The function $\zeta(s) = \sum_{n=1}^{\infty} 1/n^s$ — the Riemann zeta function — takes this family of series and turns it into a single function of $s$. The harmonic series is the singular point where $\zeta$ has a pole.

Euler’s proof that there are infinitely many primes

The divergence of the harmonic series gives a second proof that there are infinitely many primes, different from Euclid’s:

Using the Euler product (which we’ll see later), $\zeta(1) = \prod_p 1/(1-1/p)$. If there were only finitely many primes, this product would be finite. But $\zeta(1)$ is the harmonic series, which diverges. So there must be infinitely many primes. ∎

This is a fundamentally different style of proof from Euclid’s — it uses analysis (the divergence of a series) to prove a fact about number theory (the infinitude of primes). It’s a preview of the deep connections between primes and analysis that the zeta function reveals.