Calculus How To

Harmonic Series

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The harmonic series is defined as:
harmonic series

Each term of the series, except the first, is the harmonic mean of its neighbors.

The harmonic series is widely used in calculus and physics. It is a special case of the p-series, which has the form:

When p = 1, the p-series becomes the harmonic series.

Divergence of the Harmonic Series

The harmonic series diverges . This seems strange, considering the terms eventually get smaller and smaller, diminishing to zero. However, you can prove in a few different ways that is does in fact, diverge.

First, the partial sums grow without limit. That said, it takes a very long time for the sequence to grow: it takes in excess of 1043 terms to reach a sum of 100 (Thompson & Gardner, 2014).

One proof was first formulated by Nicole Oresme (1323–1382). The proof, which you can still find in textbooks today, involves grouping terms as follows:
grouping proof for the harmonic series

Each group has 1, 1, 2, 4, 8, 16… terms, and the sum of each group is at least ½. Therefore, this shows that the series diverges.

The integral test can be used to show divergence.

History of the Harmonic Series

Mathematicians developed the series based on musical notes: terms in the series were developed as fractions of the fundamental frequency in music (the lowest resonant frequency of a musical instrument). For example, ½ is twice the fundamental frequency and ⅓ is three times the fundamental frequency. The origins of the harmonic series go back as far as Pythagoras, who studied music as an abstract science (Larson & Edwards, 2008).


Chung, K. 8. Fundamental frequency and harmonics.
Larson, R. & Edwards, B. (2008). Calculus of a Single Variable. Cengage Learning.
Thompson,S. & Gardner, M. (2014). Calculus Made Easy. St. Martin’s Publishing Group.
The Harmonic Divergence. (2007). MAA Minicourse, San Jose MathFest. Retrieved February 5, 2020 from:

Stephanie Glen. "Harmonic Series" From Calculus for the rest of us!

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