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The sum is taken over all combinations of nonnegative integer indices k 1 through k m such that the sum of all k i is n. That is, for each term in the expansion, the exponents of the x i must add up to n. [1] [a] In the case m = 2, this statement reduces to that of the binomial theorem. [1]
If α is a nonnegative integer n, then all terms with k > n are zero, [5] and the infinite series becomes a finite sum, thereby recovering the binomial formula. However, for other values of α , including negative integers and rational numbers, the series is really infinite.
An infinite series of any rational function of can be reduced to a finite series of polygamma functions, by use of partial fraction decomposition, [8] as explained here. This fact can also be applied to finite series of rational functions, allowing the result to be computed in constant time even when the series contains a large number of terms.
A slight generalization of central binomial coefficients is to take them as (+) (+) = (+,), with appropriate real numbers n, where () is the gamma function and (,) is the beta function. The powers of two that divide the central binomial coefficients are given by Gould's sequence , whose n th element is the number of odd integers in row n of ...
In mathematics, summation is the addition of a sequence of numbers, called addends or summands; the result is their sum or total.Beside numbers, other types of values can be summed as well: functions, vectors, matrices, polynomials and, in general, elements of any type of mathematical objects on which an operation denoted "+" is defined.
The Basel problem is a problem in mathematical analysis with relevance to number theory, concerning an infinite sum of inverse squares.It was first posed by Pietro Mengoli in 1650 and solved by Leonhard Euler in 1734, [1] and read on 5 December 1735 in The Saint Petersburg Academy of Sciences. [2]
In mathematics, the Bernoulli numbers B n are a sequence of rational numbers which occur frequently in analysis.The Bernoulli numbers appear in (and can be defined by) the Taylor series expansions of the tangent and hyperbolic tangent functions, in Faulhaber's formula for the sum of m-th powers of the first n positive integers, in the Euler–Maclaurin formula, and in expressions for certain ...
where f is a cutoff function with appropriate properties. The cutoff function must be normalized to f(0) = 1; this is a different normalization from the one used in differential equations. The cutoff function should have enough bounded derivatives to smooth out the wrinkles in the series, and it should decay to 0 faster than the series grows.