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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 ...
The period of 1 / p 2 is usually pT p, where T p is the period of 1 / p . There are three known primes for which this is not true, and for those the period of 1 / p 2 is the same as the period of 1 / p because p 2 divides 10 p−1 −1. These three primes are 3, 487, and 56598313 (sequence A045616 in the OEIS). [11]
Continued fraction. A finite regular continued fraction, where is a non-negative integer, is an integer, and is a positive integer, for . 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 ...
In mathematics, the Leibniz formula for π, named after Gottfried Wilhelm Leibniz, states that. an alternating series . It is sometimes called the Madhava–Leibniz series as it was first discovered by the Indian mathematician Madhava of Sangamagrama or his followers in the 14th–15th century (see Madhava series ), [1] and was later ...
The convergence of the geometric series with r=1/2 and a=1/2 The convergence of the geometric series with r=1/2 and a=1 Close-up view of the cumulative sum of functions within the range -1 < r < -0.5 as the first 11 terms of the geometric series 1 + r + r 2 + r 3 + ... are added. The geometric series 1 / (1 - r) is the red dashed line.
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 ...
Division by zero. The reciprocal function y = 1 x. As x approaches zero from the right, y tends to positive infinity. As x approaches zero from the left, y tends to negative infinity. In mathematics, division by zero, division where the divisor (denominator) is zero, is a unique and problematic special case.
Partial fraction decomposition. In algebra, the partial fraction decomposition or partial fraction expansion of a rational fraction (that is, a fraction such that the numerator and the denominator are both polynomials) is an operation that consists of expressing the fraction as a sum of a polynomial (possibly zero) and one or several fractions ...