A harmonic series is the series of numbers . The harmonic series is important because it lies between series that converge and series that diverge. The harmonic series is divergent. This is especially interesting in light of the fact that the sequence of terms converges to zero. The harmonic sequence is the sequence . The harmonic sequence is convergent, but the harmonic series is divergent.
The harmonic series is so named because it contains the overtones of a vibrating strings. If a string vibrates at a particular frequency, The first overtone has a frequency of 1/2 of the original, the second 1/3, and so on.
The divergence of the harmonic series was first proved in the 14^{th} century by Nicole Oresme. Nicole Oresme used a comparison proof. A modern rendition of this proof is as follows:
Let H denote the harmonic series.
Group the terms of H as follows: Define a new series I (which will be easier to analyze by replacing each term in a bracketed grouping by the smallest term in that grouping. Clearly H is not less than the new series I, as the sum of terms in each bracketed grouping of H is not less than the sum of terms for the corresponding grouping of I.Observe that
which diverges.
is the same as
But H is not less than I, so H also diverges.
The alternating harmonic series is the series . The alternating harmonic series is convergent. This can be shown using the alternating series test.
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