The harmonic mean  of  numbers  (where , ..., ) is the number  defined by

(1)

The harmonic mean of a list of numbers may be computed in the Wolfram Language using HarmonicMean[list].

The special cases of  and  are therefore given by

			(2)
			(3)

and so on.

The harmonic means of the integers from 1 to  for , 2, ... are 1, 4/3, 18/11, 48/25, 300/137, 120/49, 980/363, ... (OEIS A102928 and A001008).

For , the harmonic mean is related to the arithmetic mean  and geometric mean  by

(4)

(Havil 2003, p. 120).

The harmonic mean is the special case  of the power mean and is one of the Pythagorean means. In older literature, it is sometimes called the subcontrary mean.

The volume-to-surface area ratio for a cylindrical container with height  and radius  and the mean curvature of a general surface are related to the harmonic mean.

Hoehn and Niven (1985) show that

(5)

for any positive constant .

REFERENCES:

Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 10, 1972.

Havil, J. Gamma: Exploring Euler's Constant. Princeton, NJ: Princeton University Press, pp. 119-121, 2003.

Hoehn, L. and Niven, I. "Averages on the Move." Math. Mag. 58, 151-156, 1985.

Kenney, J. F. and Keeping, E. S. "Harmonic Mean." §4.13 in Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, pp. 57-58, 1962.

Sloane, N. J. A. Sequences A001008/M2885 and A102928 in "The On-Line Encyclopedia of Integer Sequences."

Zwillinger, D. (Ed.). CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, p. 602, 1995.