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Binomial theorem
An ubiquitous theorem that is, in particular, often encountered in Probability theory.
The "ordinary" binomial theorem states that if n is a non negative integer, then for any x :
where
is the number of ways i objects can be selected among a collection of n objects.
The expansion of (1 + x)n has (n + 1) terms.
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Recall that the probability mass function of the binomial distribution is :
hence the name of the distribution.
It is remarkable that this result can be generalized to the case where the exponent n is in fact any real number a, provided that |x| < 1. The "generalized binomial theorem", or "binomial series theorem" states that then :
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where now, by definition :
Note that :
1) If a is not a non-negative, integer, the series does have an infinite number of terms.
2) But if a is an integer n, all the terms after the first (n + 1) terms are equal to 0. Besides, then
and we are back to the ordinary binomial theorem.
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The demonstration of the generalized binomial theorem is beyond the bounds of this Glossary.
The generalized binomial theorem can take several specialized forms. We give two of these forms, that will be used througout this site.
We want the expansion of
with n a positive integer.
We have
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Note that although n is an integer, the expansion has now an infinite number of terms.
Consider the case where x is a negative real number larger than -1. We change the notation, and want the expansion of
with both x and n positive.
We change x into -x in the above result to obtain
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In particular, if n = 1, we obtain as a special case the sum of the infinite geometric series
with |x| < 1.
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