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The gamma function is an important special function in mathematics. Its particular values can be expressed in closed form for integer and half-integer arguments, but no simple expressions are known for the values at rational points in general. Other fractional arguments can be approximated through efficient infinite products, infinite series ...
The harmonic series is the infinite series in which the terms are all of the positive unit fractions. It is a divergent series: as more terms of the series are included in partial sums of the series, the values of these partial sums grow arbitrarily large, beyond any finite limit.
For example, the parity cycle (1 0 1 1 0 0 1) has length 7 and four odd terms at indices 0, 2, 3, and 6. It is repeatedly generated by the fraction as the latter leads to the rational cycle
The geometric series 1/4 + 1/16 + 1/64 + 1/256 + ... shown as areas of purple squares. Each of the purple squares has 1/4 of the area of the next larger square (1/2×1/2= 1/4, 1/4×1/4 = 1/16, etc.). The sum of the areas of the purple squares is one third of the area of the large square. Another geometric series (coefficient a= 4/9 and common ratio r= 1/9) shown as areas of purple squares. The ...
In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities and with , is in a golden ratio to if. φ. where the Greek letter phi ( or ) denotes the golden ratio.
In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer part and another reciprocal, and so on. [1] In a finite continued fraction (or terminated continued ...
Calculus, mathematical analysis, statistics, physics. In mathematics, the gamma function (represented by Γ, the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except the non-positive integers.
[3] [4] Consider the sequence of mixed numbers with . To calculate a Pythagorean triple, take any term of this sequence and convert it to an improper fraction (for mixed number , the corresponding improper fraction is ). Then its numerator and denominator are the sides, b and a, of a right triangle, and the hypotenuse is b + 1. For example: