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1/2 + 1/4 + 1/8 + 1/16 + ⋯. First six summands drawn as portions of a square. The geometric series on the real line. In mathematics, the infinite series 1 2 + 1 4 + 1 8 + 1 16 + ··· is an elementary example of a geometric series that converges absolutely. The sum of the series is 1. In summation notation ...
e. In mathematics, a permutation group is a group G whose elements are permutations of a given set M and whose group operation is the composition of permutations in G (which are thought of as bijective functions from the set M to itself). The group of all permutations of a set M is the symmetric group of M, often written as Sym (M). [1]
In mathematics, a Fermat number, named after Pierre de Fermat (1607–1665), the first known to have studied them, is a positive integer of the form: where n is a non-negative integer. The first few Fermat numbers are: 3, 5, 17, 257, 65537, 4294967297, 18446744073709551617, ... (sequence A000215 in the OEIS). If 2 k + 1 is prime and k > 0, then ...
That is, and in accordance with Mihăilescu's theorem, the equation 2 m − 1 = n k has no solutions where m, n, and k are integers with m > 1 and k > 1. The Mersenne number sequence is a member of the family of Lucas sequences. It is U n (3, 2). That is, Mersenne number m n = 3m n-1 - 2m n-2 with m 0 = 0 and m 1 = 1.
In number theory, primes in arithmetic progression are any sequence of at least three prime numbers that are consecutive terms in an arithmetic progression. An example is the sequence of primes (3, 7, 11), which is given by for . According to the Green–Tao theorem, there exist arbitrarily long arithmetic progressions in the sequence of primes.
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 first few terms of the sequence are 1, 1, 3, 7, 17, 41, 99, … (sequence A001333 in the OEIS). Each term in this sequence is half the corresponding term in the sequence of companion Pell numbers. These numbers also appear in the continued fraction convergents to √ 2. Further reading. Newman, M.; Shanks, D. & Williams, H. C. (1980).
In combinatorics, the Eulerian number is the number of permutations of the numbers 1 to in which exactly elements are greater than the previous element (permutations with "ascents"). Leonhard Euler investigated them and associated polynomials in his 1755 book Institutiones calculi differentialis. The polynomials presently known as Eulerian ...
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