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A harmonic series is a series that contains the sum of terms that are the reciprocals of an arithmetic series’ terms. This article will explore this unique series and understand how they behave as an infinite series.
The harmonic series is the infinite series in which the terms are all of the positive unit fractions. It is a divergent series: as more terms of the series are included in partial sums of the series, the values of these partial sums grow arbitrarily large, beyond any finite limit.
The series sum_(k=1)^infty1/k (1) is called the harmonic series. It can be shown to diverge using the integral test by comparison with the function 1/x. The divergence, however, is very slow.
Briefly, the harmonic series, also referred to as the overtone series, occurs whenever you play a pitch in your instrument. When you play a C, what you are hearing is a collection of overtones associated to this pitch and this is applicable to any sound you hear coming from an instrument or otherwise.
We say that these series converge (meaning that they can be assigned a finite value). This section is concerned with another very natural series, the so-called harmonic series. 1 1 + 1 2 + 1 3 + ⋯ + 1 n + ⋯ (for ever). It is not entirely clear why this is called the harmonic series.
Definition. The harmonic series is the infinite series formed by the sum of the reciprocals of the natural numbers, expressed mathematically as $$ ext {H} = 1 + frac {1} {2} + frac {1} {3} + frac {1} {4} + ext {...}$$.
This calculus 2 video provides a basic introduction into the harmonic series. It explains why the harmonic series diverges using the integral test for serie...