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For positive values of a and b, the binomial theorem with n = 2 is the geometrically evident fact that a square of side a + b can be cut into a square of side a, a square of side b, and two rectangles with sides a and b.
The binomial coefficients can be arranged to form Pascal's triangle, in which each entry is the sum of the two immediately above. Visualisation of binomial expansion up to the 4th power. In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem.
For larger positive integer values of n, the correct result is given by the binomial theorem. The name "freshman's dream" also sometimes refers to the theorem that says that for a prime number p, if x and y are members of a commutative ring of characteristic p, then (x + y) p = x p + y p.
The approximation can be proven several ways, and is closely related to the binomial theorem. By Bernoulli's inequality , the left-hand side of the approximation is greater than or equal to the right-hand side whenever x > − 1 {\displaystyle x>-1} and α ≥ 1 {\displaystyle \alpha \geq 1} .
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. There is no restriction on the relative sizes of n and k , [ 1 ] since, if n < k the value of the binomial coefficient is zero and the identity remains valid.
The value of each is taken to be 1 ... is a positive integer, () ... where the coefficients are the same as those in the binomial theorem.
The Leibniz rule bears a strong resemblance to the binomial theorem, and in fact the binomial theorem can be proven directly from the Leibniz rule by taking () = and () =, which gives ( a + b ) n e ( a + b ) x = e ( a + b ) x ∑ k = 0 n ( n k ) a n − k b k , {\displaystyle (a+b)^{n}e^{(a+b)x}=e^{(a+b)x}\sum _{k=0}^{n}{\binom {n}{k}}a^{n-k}b ...
Faulhaber's formula concerns expressing the sum of the p-th powers of the first n positive integers = = + + + + as a (p + 1)th-degree polynomial function of n.. The first few examples are well known.