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The harmonic number with = ⌊ ⌋ (red line) with its asymptotic limit + (blue line) where is the Euler–Mascheroni constant.. In mathematics, the n-th harmonic number is the sum of the reciprocals of the first n natural numbers: [1] = + + + + = =.
A graphical interpretation of the harmonic mean, z of two numbers, x and y, and a nomogram to calculate it. The blue line shows that the harmonic mean of 6 and 2 is 3. The magenta line shows that the harmonic mean of 6 and −2 is −6.
The block-stacking problem: blocks aligned according to the harmonic series can overhang the edge of a table by the harmonic numbers In the block-stacking problem , one must place a pile of n {\displaystyle n} identical rectangular blocks, one per layer, so that they hang as far as possible over the edge of a table without falling.
2.6 Harmonic numbers. 3 Binomial coefficients. ... The following is a useful property to calculate low-integer-order polylogarithms recursively in closed form:
Equivalently, a sequence is a harmonic progression when each term is the harmonic mean of the neighboring terms. As a third equivalent characterization, it is an infinite sequence of the form 1 a , 1 a + d , 1 a + 2 d , 1 a + 3 d , ⋯ , {\displaystyle {\frac {1}{a}},\ {\frac {1}{a+d}},\ {\frac {1}{a+2d}},\ {\frac {1}{a+3d}},\cdots ,}
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The integral is Euler's harmonic number, so the previous formula ... For x < 1, we can calculate limits based on the fact that between 1 and 2, ...
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