The harmonic series, the "prime harmonic series", and their difference

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Harmonic series Consider the harmonic series 1 + 1 2 + 1 3 + … 1 + \frac12 + \frac13 + \dots 1 + 2 1 ​ + 3 1 ​ + … . Apparently, it was Donald Knuth who in 1968 (vol 1 of TAOCP, section 1.2.7 “Harmonic numbers”) gave the name “harmonic number” and the notation H n H_n H n ​ to the partial sums of this series: H n = 1 + 1 2 + 1 3 + ⋯ + 1 n H_n = 1 + \frac12 + \frac13 + \dots…

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