Harmonic series

Definition

The harmonic series is the infinite series given by the sum of $\sum_{n=1}^{\infty} \frac{1}{n}$. It is a divergent series, meaning its partial sums grow without bound.

5 Must Know Facts For Your Next Test

1. The harmonic series diverges, even though the terms of the series approach zero.
2. The $n^{th}$ partial sum of the harmonic series can be approximated by $\ln(n) + \gamma$ where $\gamma$ is the Euler-Mascheroni constant.
3. The divergence of the harmonic series can be shown using the integral test or comparison test.
4. Although it diverges, if you take every other term, forming a new series like $\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n}$, this new alternating harmonic series converges to $\ln(2)$.
5. Despite divergence, harmonic numbers (partial sums of harmonic series) appear in various applications such as number theory and computer science.

Review Questions

• What is the general form of the terms in a harmonic series?
• Does the harmonic series converge or diverge? Explain why.
• How can you approximate the $n^{th}$ partial sum of the harmonic series?

Related terms

Geometric Series: An infinite series where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.

Alternating Series: A series whose terms are alternately positive and negative. A famous example is the alternating harmonic series which converges.

Integral Test: $$A method used to determine whether an infinite series converges or diverges by comparing it to an improper integral.$$