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.
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The harmonic series diverges, even though the terms of the series approach zero.
The $n^{th}$ partial sum of the harmonic series can be approximated by $\ln(n) + \gamma$ where $\gamma$ is the Euler-Mascheroni constant.
The divergence of the harmonic series can be shown using the integral test or comparison test.
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 $\ln(2)$.
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: $$A method used to determine whether an infinite series converges or diverges by comparing it to an improper integral.$$