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[1] [2] Every term of the harmonic series after the first is the harmonic mean of the neighboring terms, so the terms form a harmonic progression; the phrases harmonic mean and harmonic progression likewise derive from music. [2] Beyond music, harmonic sequences have also had a certain popularity with architects.
The geometric series 1 / 2 − 1 / 4 + 1 / 8 − 1 / 16 + ⋯ sums to 1 / 3 .. The alternating harmonic series has a finite sum but the harmonic series does not.
An infinite series of any rational function of can be reduced to a finite series of polygamma functions, by use of partial fraction decomposition, [8] as explained here. This fact can also be applied to finite series of rational functions, allowing the result to be computed in constant time even when the series contains a large number of terms.
A classic example is the alternating harmonic series given by + + = = +, which converges to (), but is not absolutely convergent (see Harmonic series). Bernhard Riemann proved that a conditionally convergent series may be rearranged to converge to any value at all, including ∞ or −∞; see Riemann series theorem .
In a third layer, the logarithms of rational numbers r = a / b are computed with ln(r) = ln(a) − ln(b), and logarithms of roots via ln n √ c = 1 / n ln(c).. The logarithm of 2 is useful in the sense that the powers of 2 are rather densely distributed; finding powers 2 i close to powers b j of other numbers b is comparatively easy, and series representations of ln(b) are ...
The alternating harmonic series is a classic example of a conditionally convergent series: = + is convergent, whereas = | + | = = is the ordinary harmonic series, which diverges. Although in standard presentation the alternating harmonic series converges to ln(2) , its terms can be arranged to converge to any number, or even to diverge.
In mathematical analysis, the alternating series test proves that an alternating series is convergent when its terms decrease monotonically in absolute value and approach zero in the limit. The test was devised by Gottfried Leibniz and is sometimes known as Leibniz's test , Leibniz's rule , or the Leibniz criterion .
An example of a conditionally convergent series is the alternating harmonic series. Many standard tests for divergence and convergence, most notably including the ratio test and the root test, demonstrate absolute convergence. This is because a power series is absolutely convergent on the interior of its disk of convergence. [a]