#MathsInMinutes Day 47: The harmonic series

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#MathsInMinutes Day 47: The harmonic series

The harmonic series is the sum of an infinite sequence of steadily diminishing fractions. Important to the theory of music, it is defined as:

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One surprising aspect of the harmonic series is that it grows without bounds even though the successive differences between its terms shrink towards zero.

One way of spotting this divergent behavior is to collect the terms together in smaller groups. This reveals that it is always possible to form a group of successively smaller terms that, together, sum to a number larger than one half. For instance, (1/3 + ¼ ) is greater than one half, as is (1/5 + 1/6 + 1/7 + 1/8).

The harmonic series is important in music because it gives the various modes of vibration for a plucked or struck string that is fixed at both ends

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