Sage-Tips

Is the harmonic series ever an integer?

Let Hn be the nth harmonic number. Then Hn is not an integer for n≥2. That is, the only harmonic numbers that are integers are H0 and H1.

How are harmonics numbered?

Harmonic numbers are related to the harmonic mean in that the n-th harmonic number is also n times the reciprocal of the harmonic mean of the first n positive integers. The harmonic numbers roughly approximate the natural logarithm function and thus the associated harmonic series grows without limit, albeit slowly.

What is the harmonic sum of 5?

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n	H(n)	≈H(n)
4	25/12	2.08333
5	137/60	2.28333
6	49/20	2.45
7	363/140	2.59286

What is nth harmonic number?

The harmonic numbers are the partial sums of the harmonic series. The n th n^\text{th} nth harmonic number is the sum of the reciprocals of each positive integer up to n.

Do harmonics always diverge?

By the limit comparison test with the harmonic series, all general harmonic series also diverge.

What is the 0th harmonic?

0 is cos(0 t) – the d.c. component, which I would call the 0th harmonic. The first coefficient (non-zero) is the fundamental, the first harmonic.

What are harmonic prime numbers?

Prime harmonic series

N (primes)	scale
8 (1,3,5,7,11,13,17,19)	1/1, 17/16, 19/16, 5/4, 11/8, 3/2, 13/8, 7/4 (octatonic)
9 (1,3,5,7,11,13,17,19,23)	1/1, 17/16, 19/16, 5/4, 11/8, 23/16, 3/2, 13/8, 7/4 (nonotonic)
10 (1,3,5,7,11,13,17,19,23,29)	1/1, 17/16, 19/16, 5/4, 11/8, 23/16, 3/2, 13/8, 7/4, 29/16 (decatonic)