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In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial.According to the theorem, the power (+) expands into a polynomial with terms of the form , where the exponents and are nonnegative integers satisfying + = and the coefficient of each term is a specific positive integer ...
The binomial approximation is useful for approximately calculating powers of sums of 1 and a small number x. It states that It states that ( 1 + x ) α ≈ 1 + α x . {\displaystyle (1+x)^{\alpha }\approx 1+\alpha x.}
Differentiating term-wise the binomial series within the disk of convergence | x | < 1 and using formula , one has that the sum of the series is an analytic function solving the ordinary differential equation (1 + x)u′(x) − αu(x) = 0 with initial condition u(0) = 1. The unique solution of this problem is the function u(x) = (1 + x) α.
For natural numbers (taken to include 0) n and k, the binomial coefficient () can be defined as the coefficient of the monomial X k in the expansion of (1 + X) n. The same coefficient also occurs (if k ≤ n) in the binomial formula
This list of mathematical series contains formulae for finite and infinite sums. It can be used in conjunction with other tools for evaluating sums. Here, is taken to have the value
The Taylor polynomials for ln(1 + x) only provide accurate approximations in the range −1 < x ≤ 1. For x > 1, Taylor polynomials of higher degree provide worse approximations. The Taylor approximations for ln(1 + x) (black). For x > 1, the approximations diverge. Pictured is an accurate approximation of sin x around the point x = 0. The ...
In mathematics, the Bernoulli polynomials, named after Jacob Bernoulli, combine the Bernoulli numbers and binomial coefficients. They are used for series expansion of functions, and with the Euler–MacLaurin formula. These polynomials occur in the study of many special functions and, in particular, the Riemann zeta function and the Hurwitz ...
In mathematics, Pascal's rule (or Pascal's formula) is a combinatorial identity about binomial coefficients.It states that for positive natural numbers n and k, + = (), where () is a binomial coefficient; one interpretation of the coefficient of the x k term in the expansion of (1 + x) n.