Harmonic Series

Summary

The harmonic series $\sum 1/n$ diverges, even though its terms tend to zero. Partial sums grow like $\ln n+\gamma$ . Rearrangement theorems apply to the alternating harmonic series, not to the divergent ordinary harmonic series.

Prerequisites

Series Sums by Partial Sums, Integral Test

Definition

The $n$ th harmonic number is $H_n=\sum _{k=1}^{n}1/k$ .

Theorem

The harmonic series diverges. One proof uses the integral test:

so $\sum 1/n$ diverges. Grouping also shows $H_{2^m}\ge 1+m/2\to \mathrm{\infty }$ .

Asymptotically,

where $\gamma \approx 0.57721$ is the Euler–Mascheroni constant.

Worked Example

$H_5=1+1/2+1/3+1/4+1/5=137/60$ .

The alternating harmonic series $\sum (-1{)}^{n+1}/n$ converges (to $\ln 2$ ) and is conditionally convergent; its rearrangements can change the sum. The ordinary harmonic series does not converge, so it has no finite sum to rearrange.

Common Mistakes

• Believing $a_n\to 0$ implies convergence (false; harmonic is the classic counterexample).
• Attributing Riemann’s rearrangement theorem to $\sum 1/n$ itself.

Connections

• P Series ( $p=1$ boundary case)
• Alternating Series, Absolute Convergence

References

Divergence of the harmonic series is standard in OpenStax Calculus Volume 2.^[1]