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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 ...
where the above convention for the coefficients of the polynomials agrees with the definition of the binomial coefficients, because both give zero for all i > m and j > n, respectively. By comparing coefficients of x r , Vandermonde's identity follows for all integers r with 0 ≤ r ≤ m + n .
As a 16-year-old, Abel gave a rigorous proof of the binomial theorem valid for all numbers, extending Euler's result which had held only for rationals. [13] [14] Abel wrote a fundamental work on the theory of elliptic integrals, containing the foundations of the theory of elliptic functions.
The following proof uses the q-binomial theorem. One standard proof of the Chu–Vandermonde identity is to expand the product (+) (+) in two different ways. Following Stanley, [1] we can tweak this proof to prove the q-Vandermonde identity, as well. First, observe that the product
In mathematics, Bertrand's postulate (now a theorem) states that, for each , there is a prime such that < <.First conjectured in 1845 by Joseph Bertrand, [1] it was first proven by Chebyshev, and a shorter but also advanced proof was given by Ramanujan.
Kash Patel, nominee to be Director of the Federal Bureau of Investigation, testifies in front of the Senate Judiciary Committee in Washington, DC on January 30, 2025.
In mathematics, Dixon's identity (or Dixon's theorem or Dixon's formula) is any of several different but closely related identities proved by A. C. Dixon, some involving finite sums of products of three binomial coefficients, and some evaluating a hypergeometric sum.
LONDON/SINGAPORE (Reuters) -The dollar drifted lower and stocks were cautiously positive on Monday as investors awaited an expected flurry of policy announcements during the first hours of Donald ...