The harmonic series — divergence

What just happened?

You kept adding ever smaller fractions. Each one is tiny, and yet the partial sum kept rising steadily — and nothing stops it. This is one of the most surprising results in analysis: even though the terms $1/n$ tend to zero, their sum tends to infinity.

The harmonic series:

The partial sum $H(N)$ grows without bound, though ever more slowly — asymptotically it behaves like the natural logarithm:

More precisely $H(N) \approx \ln N + \gamma$, where $\gamma \approx 0.5772$ is the Euler–Mascheroni constant. Since $\ln N \to \infty$ as $N \to \infty$, the sum has no ceiling — the series is divergent.

Why is this so surprising?

The mere fact that the terms shrink to zero does not guarantee that the sum is finite. Compare with the geometric series (the pink line on the scene):

Here the terms also tend to zero, but the sum is finite (convergent, equal to 1), because they shrink fast enough. In the harmonic series the terms $1/n$ shrink too slowly for the sum to stop. So the condition “the terms tend to zero” is necessary, but NOT sufficient for the series to converge.

Oresme’s proof (c. 1350):

The divergence was proved by the French scholar (bishop) Nicole Oresme in the 14th century with a brilliant method of grouping the terms: $\tfrac12 \ge \tfrac12$, $\tfrac13+\tfrac14 > \tfrac12$, $\tfrac15+\tfrac16+\tfrac17+\tfrac18 > \tfrac12$, and so on — each successive group gives more than $\tfrac12$. There are infinitely many such groups, so the sum exceeds any number set in advance: it must be infinite.

Why this matters for your exams:

The harmonic series is the textbook counterexample: it shows that $a_n \to 0$ does NOT imply convergence of the series $\sum a_n$. Remember it together with the geometric series $\tfrac12+\tfrac14+\dots=1$ as a “divergent vs convergent” pair.

Remember: $1+\tfrac12+\tfrac13+\dots$ is divergent — the sum grows like $\ln N$ and tends to infinity, even though the terms tend to zero. The terms shrinking to zero is a necessary but not sufficient condition for convergence.