The harmonic numbers are also frequently used to define ? using the limit introduced earlier:
12.
Since that sum is divergent, we get an infinite return-but in practice the harmonic numbers grows very slowly.
13.
It is known, that the harmonic numbers are never integers except the case " n = 1 ".
14.
This relation is also frequently used to define the extension of the harmonic numbers to non-integer " n ".
15.
Is there an easy ( and perhaps efficient ) way to compute the " n "-th harmonic number in PARI / GP?
16.
The analysis goes like this : If we restrict the game to k rounds, the estimated return is the sum of the first k harmonic numbers.
17.
Anyway, the point stands harmonic numbers come up quite often and quite naturally in many places in mathematics, and so they have been rather extensively studied.
18.
Harmonic numbers are related to the harmonic mean in that the-th harmonic number is also times the reciprocal of the harmonic mean of the first positive integers.
19.
Harmonic numbers are related to the harmonic mean in that the-th harmonic number is also times the reciprocal of the harmonic mean of the first positive integers.
20.
The digamma function ? appears in the formula for the differential entropy as a consequence of Euler's integral formula for the harmonic numbers which follows from the integral:
