The Binomial Theorem + Binomial Expansion + Binomial Series
By DarthVader
Date: 2022-10-17
Topic: 173 see comments
Post views: 5093
The Binomial Theorem
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The binomial theorem is used to multiply out brackets of the form; (a + b)n
In sigma notation this is:
(a + b)n = ∑nr = 0 ( n! / (n − r)!r! ) an − r br
Where:
n = The index of the binomial expression i.e (a + b)n
Binomial Expansion:
Binomial expansion is when a binomial is raised to a power i.e (a + b)n
The symbol nCr , can be used in place of a binomial coefficient i.e ( n! / (n − r)!r! ) that isn't known or hasn't been calculated yet:
(a + b)n = ∑nr = 0 nCr an − r br
Where:
nCr = Coefficient r in row n of Pascal's triangle
The Binomial Series
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If α is any real number, then the binomial series is:
(1 + x)k = [1] + [(kx) / 1!] + [(k(k − 1)x2) / 2!] + [(k(k − 1)(k − 2)x3) / 3!] + …
Where:
−1 < x < 1
Note that this is an infinite series, as indicated by the ellipses.
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