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  2. Harmonic series (mathematics) - Wikipedia

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

    Calculus. In mathematics, the harmonic series is the infinite series formed by summing all positive unit fractions : The first terms of the series sum to approximately , where is the natural logarithm and is the Euler–Mascheroni constant. Because the logarithm has arbitrarily large values, the harmonic series does not have a finite limit: it ...

  3. Stern–Brocot tree - Wikipedia

    en.wikipedia.org/wiki/Stern–Brocot_tree

    For instance, the continued fraction representation of 13 ⁄ 9 is [1;2,4] and its two children are [1;2,5] = 16 ⁄ 11 (the right child) and [1;2,3,2] = 23 ⁄ 16 (the left child). It is clear that for each finite continued fraction expression one can repeatedly move to its parent, and reach the root [1;]= 1 ⁄ 1 of the tree in finitely many ...

  4. Golden ratio - Wikipedia

    en.wikipedia.org/wiki/Golden_ratio

    The golden ratio's negative −φ and reciprocal φ−1 are the two roots of the quadratic polynomial x2 + x − 1. The golden ratio is also an algebraic number and even an algebraic integer. It has minimal polynomial. This quadratic polynomial has two roots, and. The golden ratio is also closely related to the polynomial.

  5. Repeating decimal - Wikipedia

    en.wikipedia.org/wiki/Repeating_decimal

    For example, the fraction2 / 7 ⁠ has d = 7, and the smallest k that makes 10 k − 1 divisible by 7 is k = 6, because 999999 = 7 × 142857. The period of the fraction2 / 7 ⁠ is therefore 6.

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  7. 1 − 2 + 3 − 4 + ⋯ - Wikipedia

    en.wikipedia.org/wiki/1_%E2%88%92_2_%2B_3_%E2%88...

    The idea becomes clearer by considering the general series 1 − 2x + 3x 2 − 4x 3 + 5x 4 − 6x 5 + &c. that arises while expanding the expression 1 ⁄ (1+x) 2, which this series is indeed equal to after we set x = 1.

  8. Septuple meter - Wikipedia

    en.wikipedia.org/wiki/Septuple_meter

    21. 8 as 7. 4 with 3 subdivisions. Septuple meter (British: metre) or (chiefly British) septuple time is a meter with each bar (American: measure) divided into 7 notes of equal duration, usually 7. 4 or 7. 8 (or in compound meter, 21. 8 time).

  9. Doomsday rule - Wikipedia

    en.wikipedia.org/wiki/Doomsday_rule

    Doomsday rule. The Doomsday rule, Doomsday algorithm or Doomsday method is an algorithm of determination of the day of the week for a given date. It provides a perpetual calendar because the Gregorian calendar moves in cycles of 400 years. The algorithm for mental calculation was devised by John Conway in 1973, [ 1][ 2] drawing inspiration from ...

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