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    Harmonic Sum

    Alternate names

    harmonic progression | harmonic sum

    Basic definition

    The harmonic series is the slowly divergent sum of the reciprocals of all positive integers.

    Detailed definition

    The series sum_(k = 1)^∞ 1/k is called the harmonic series. It can be shown to diverge using the integral test by comparison with the function 1/x. The divergence, however, is very slow. Divergence of the harmonic series was first demonstrated by Nicole d'Oresme (ca. 1323-1382), but was mislaid for several centuries (Havil 2003, p. 23; Derbyshire 2004, pp. 9-10). The result was proved again by Pietro Mengoli in 1647, by Johann Bernoulli in 1687, and by Jakob Bernoulli shortly thereafter.

    Related terms

    alternating harmonic series | arithmetic series | Bernoulli's paradox | book stacking problem | Euler sum | Kempner series | Madelung constants | Mertens constant | q-harmonic series | Zipf's law

    Educational grade level

    college level (AP calculus BC)

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