In mathematics, the binomial series is a generalization of the polynomial that comes from a binomial formula expression like for a nonnegative integer . Specifically, the binomial series is the MacLaurin series for the function , where and . Explicitly,
(1)
where the power series on the right-hand side of (1) is expressed in terms of the (generalized) binomial coefficients
Note that if α is a nonnegative integer n then the xn + 1 term and all later terms in the series are 0, since each contains a factor of (n − n). Thus, in this case, the series is finite and gives the algebraic binomial formula.
In mathematics, the binomialseries is a generalization of the polynomial that comes from a binomial formula expression like ( 1 + x ) n {\displaystyle...
In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial. According to the theorem...
In probability theory and statistics, the binomial distribution with parameters n and p is the discrete probability distribution of the number of successes...
mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is...
In combinatorics, the binomial transform is a sequence transformation (i.e., a transform of a sequence) that computes its forward differences. It is closely...
In probability theory and statistics, the negative binomial distribution is a discrete probability distribution that models the number of failures in a...
The binomial approximation is useful for approximately calculating powers of sums of 1 and a small number x. It states that ( 1 + x ) α ≈ 1 + α x . {\displaystyle...
filters) BinomialseriesBinomial theorem Binomial transform Binomial type Carlson's theorem Catalan number Fuss–Catalan number Central binomial coefficient...
in a series of n {\displaystyle n} independent Bernoulli trials, where each trial has probability of success p {\displaystyle p} . In binomial regression...
In mathematics the nth central binomial coefficient is the particular binomial coefficient ( 2 n n ) = ( 2 n ) ! ( n ! ) 2 = ∏ k = 1 n n + k k for all ...
statistics, a binomial proportion confidence interval is a confidence interval for the probability of success calculated from the outcome of a series of success–failure...
which the index of each polynomial equals its degree, is said to be of binomial type if it satisfies the sequence of identities p n ( x + y ) = ∑ k = 0...
In computer science, a binomial heap is a data structure that acts as a priority queue. It is an example of a mergeable heap (also called meldable heap)...
{\displaystyle {\tbinom {n}{k}}.} Like the binomial distribution that involves binomial coefficients, there is a negative binomial distribution in which the multiset...
Gaussian binomial coefficients (also called Gaussian coefficients, Gaussian polynomials, or q-binomial coefficients) are q-analogs of the binomial coefficients...
A Treatise on the Binomial Theorem is a fictional work of mathematics by the young Professor James Moriarty, the criminal mastermind and archenemy of the...
(4m+3)!}={\frac {1}{2}}\left(\sinh {z}-\sin {z}\right).} Multisection of a binomial expansion ( 1 + x ) n = ( n 0 ) x 0 + ( n 1 ) x + ( n 2 ) x 2 + ⋯ {\displaystyle...
In the theory of probability and statistics, a Bernoulli trial (or binomial trial) is a random experiment with exactly two possible outcomes, "success"...
map projection. It can be evaluated by expanding the integral by the binomialseries and integrating term by term: see Meridian arc for details. The length...
either by composition with the binomialseries (1+x)α, or by composition with the exponential and the logarithmic series, f α = exp ( α log ( f ) )...
{63}{256}}\beta ^{10}+\cdots ,\end{aligned}}} which is a special case of a binomialseries. The approximation γ ≈ 1 + 1 2 β 2 {\textstyle \gamma \approx 1+{\frac...
The binomial system (Spanish: Sistema binominal) is a voting system that was used in the legislative elections of Chile between 1989 and 2013. From an...
of the terms in that sum. It is the generalization of the binomial theorem from binomials to multinomials. For any positive integer m and any non-negative...
^{4}}{8z^{3}}}+\cdots \end{aligned}}} If we consider all the terms of binomialseries, then there is no approximation. Let us substitute this expression...
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