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2.4 The Harmonic SeriesThe recognition that there is a connection between the area under a hyperbola and the logarithm came in the middle of the seventeenth century. The Belgian Jesuit Gregorius Saint Vincent (1584–1667) observed in his Opus Geometricum of 1647 that for any constant m, the area between the hyperbola xy = 1 and the xaxis from x = a to x = b is the same as the area from x = ma to x = mb. In modern notation, this says that (2.4.1) Two years later, in Solutio Problematis a Mersenno Proposit, his student Alfonzo Antonio de Sarasa (1618–1667) showed that if we define f(x) by then Gregory's identity implies that (2.4.2) This relationship and the fact that f is strictly increasing defines f as a logarithm. In the case of Sarasa's function, the base of this logarithm is the number—denoted by e—that satisfies Because of the pivotal role it has come to play in calculus, this logarithm is now called the natural logarithm and is denoted by Around 1667, Newton observed that the geometric series could be used to find the power series expansion of ln(1+x): Nicolaus
Mercator (1620–1687) found
the same series at about the same time and was the first to publish
it. Like
the geometric
and binomial series, it is not valid for all values of x.
It holds when For x =
1,
it yields the important identity If we try setting x = –1 in equation (2.4.3), we get nonsense that nevertheless suggests something very meaningful: The series in parentheses is the harmonic series, so called because it is the sum of the harmonic ratios—those ratios of frequencies with one in the numerator and a small integer in the denominator that represent the fundamental harmonies. 
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