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It is known that ζ(3) is irrational (Apéry's theorem) and that infinitely many of the numbers ζ(2n + 1) : n ∈ , are irrational. There are also results on the irrationality of values of the Riemann zeta function at the elements of certain subsets of the positive odd integers; for example, at least one of ζ (5), ζ (7), ζ (9), or ζ (11 ...
Method of image charges. The method of image charges (also known as the method of images and method of mirror charges) is a basic problem-solving tool in electrostatics. The name originates from the replacement of certain elements in the original layout with fictitious charges, which replicates the boundary conditions of the problem (see ...
Negative refraction. Negative refraction is the electromagnetic phenomenon where light rays become refracted at an interface that is opposite to their more commonly observed positive refractive properties. Negative refraction can be obtained by using a metamaterial which has been designed to achieve a negative value for (electric) permittivity ...
The partial sums of the series 1 + 2 + 3 + 4 + 5 + 6 + ⋯ are 1, 3, 6, 10, 15, etc.The nth partial sum is given by a simple formula: = = (+). This equation was known ...
Moseley's law. Moseley's law is an empirical law concerning the characteristic X-rays emitted by atoms. The law had been discovered and published by the English physicist Henry Moseley in 1913–1914. [1] [2] Until Moseley's work, "atomic number" was merely an element's place in the periodic table and was not known to be associated with any ...
In physics, a charged particle is a particle with an electric charge. For example, some elementary particles, like the electron or quarks are charged. [1] Some composite particles like protons are charged particles. An ion, such as a molecule or atom with a surplus or deficit of electrons relative to protons are also charged particles.
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.
Lucas numbers have L 1 = 1, L 2 = 3, and L n = L n−1 + L n−2. Primefree sequences use the Fibonacci recursion with other starting points to generate sequences in which all numbers are composite. Letting a number be a linear function (other than the sum) of the 2 preceding numbers. The Pell numbers have P n = 2P n−1 + P n−2.