Harmonic Series¶

Definition¶

The harmonic series is an infinite sum of the reciprocals of positive integers, i.e.,

\displaystyle\sum_{n=1}^\infty \frac{1}{n}\text{.}

Interpretation¶

The harmonic series is the standard example of an infinite series whose terms tend to zero but whose sum diverges. It shows that vanishing terms are necessary for convergence, but not sufficient.

The partial sums of the harmonic series for n up to 30 growing without bound alongside the curve ln n plus gamma, while the terms 1/n shrink towards 0.

The terms $1/n$ shrink towards $0$ , but the partial sums $H_n$ keep climbing, close to the dashed curve $\ln n+\gamma$ with $\gamma\approx0.5772$ $0.5772$ . The growth is slow, $H_{30}\approx3.99$ and $H_{1000}\approx7.49$ $7.49$ , but there is no upper bound, so $H_n$ diverges to infinity. The grouping argument in the Examples proves this: every block of terms between consecutive powers of two adds at least $\tfrac{1}{2}$ .

A common slip is to conclude from $1/n\to0$ $0$ , or from partial sums that seem to level off in a computation, that the series converges. The condition that the terms tend to $0$ can only rule convergence out, never prove it.

The harmonic series is the benchmark on the divergent side: by the comparison test, any series with nonnegative terms at least $1/n$ , such as $\displaystyle \sum 1/\sqrt{n}$ , diverges as well. With alternating signs the picture changes: $1-\tfrac{1}{2}+\tfrac{1}{3}-\cdots$ converges by the alternating series test, to $\ln2$ . The harmonic series returns as the borderline case $p=1$ of the $p$ -series $\displaystyle \sum 1/n^p$ .

Key Results¶

The harmonic series diverges to infinity.
Divergence occurs despite terms decreasing to zero.

Examples¶

Group the terms of the partial sum $\displaystyle H_{2^m}=\sum_{n=1}^{2^m}1/n$ into blocks ending at powers of two. For each $j=1,\ldots,m$ $1, \dots, m$ , the block from $n=2^{j-1}+1$ to $n=2^j$ contains $2^{j-1}$ terms, each at least $1/2^j$ . Thus every block contributes at least $1/2$ , and

H_{2^m}\ge 1+\frac{m}{2}\longrightarrow+\infty.

The partial sums are increasing, so the whole sequence of partial sums diverges to infinity, even though $1/n\to0$ $0$ .

Prerequisites¶

Sources¶

Math I: Calculus · Lecture 10

Symbols: gamma, infinity, square root, sum

Last updated: 20 October 2026 03:02