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  2. List of sums of reciprocals - Wikipedia

    en.wikipedia.org/wiki/List_of_sums_of_reciprocals

    The sum of the reciprocals of the powerful numbers is close to 1.9436 . [4] The reciprocals of the factorials sum to the transcendental number e (one of two constants called "Euler's number"). The sum of the reciprocals of the square numbers (the Basel problem) is the transcendental number ⁠ π 2 / 6 ⁠, or ζ(2) where ζ is the Riemann zeta ...

  3. Harmonic progression (mathematics) - Wikipedia

    en.wikipedia.org/wiki/Harmonic_progression...

    In mathematics, a harmonic progression (or harmonic sequence) is a progression formed by taking the reciprocals of an arithmetic progression, which is also known as an arithmetic sequence. Equivalently, a sequence is a harmonic progression when each term is the harmonic mean of the neighboring terms.

  4. Harmonic number - Wikipedia

    en.wikipedia.org/wiki/Harmonic_number

    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] = + + + + = =.

  5. Harmonic series (mathematics) - Wikipedia

    en.wikipedia.org/wiki/Harmonic_series_(mathematics)

    [8] [9] [10] Bernoulli credited his brother Johann Bernoulli for finding the proof, [10] and it was later included in Johann Bernoulli's collected works. [11] The partial sums of the harmonic series were named harmonic numbers, and given their usual notation , in 1968 by Donald Knuth. [12]

  6. Harmonic mean - Wikipedia

    en.wikipedia.org/wiki/Harmonic_mean

    The harmonic mean is the reciprocal of the arithmetic mean of the reciprocals of the numbers, that is, the generalized f-mean with () =. For example, the harmonic mean of 1, 4, and 4 is For example, the harmonic mean of 1, 4, and 4 is

  7. Divergence of the sum of the reciprocals of the primes

    en.wikipedia.org/wiki/Divergence_of_the_sum_of...

    While the partial sums of the reciprocals of the primes eventually exceed any integer value, they never equal an integer. One proof [6] is by induction: The first partial sum is ⁠ 1 / 2 ⁠, which has the form ⁠ odd / even ⁠. If the n th partial sum (for n ≥ 1) has the form ⁠ odd / even ⁠, then the (n + 1) st sum is

  8. Reciprocals of primes - Wikipedia

    en.wikipedia.org/wiki/Reciprocals_of_primes

    A prime p (where p ≠ 2, 5 when working in base 10) is called unique if there is no other prime q such that the period length of the decimal expansion of its reciprocal, 1/p, is equal to the period length of the reciprocal of q, 1/q. [8]

  9. Reciprocal - Wikipedia

    en.wikipedia.org/wiki/Reciprocal

    Reciprocal polynomial, a polynomial obtained from another polynomial by reversing its coefficients; Reciprocal rule, a technique in calculus for calculating derivatives of reciprocal functions; Reciprocal spiral, a plane curve; Reciprocal averaging, a statistical technique for aggregating categorical data