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binomial formula
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(Theorem)
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The binomial formula gives the power series expansion of the
 power function. The power  can be an integer, rational, real, or even a complex number. The formula is
where
 denotes the falling factorial, and where
 denotes the generalized binomial coefficient.
For
 the power series reduces to a polynomial, and we obtain the usual binomial theorem. For other values of  , the radius of convergence of the series is  ; the right-hand series converges pointwise for all complex  to the value on the left side. Also note that the binomial formula is valid at  , but for certain values of  only. Of course, we have convergence if  is a natural number. Furthermore, for  and real  , we have absolute convergence if  , and conditional convergence if  . For  we have absolute convergence for  .
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"binomial formula" is owned by rmilson.
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Cross-references: conditional convergence, absolute convergence, natural number, side, complex, pointwise, converges, series, radius of convergence, binomial theorem, polynomial, generalized binomial coefficient, falling factorial, complex number, even, real, rational, integer, power, power function, power series
There are 11 references to this entry.
This is version 8 of binomial formula, born on 2002-02-19, modified 2006-03-19.
Object id is 2204, canonical name is BinomialFormula.
Accessed 26910 times total.
Classification:
| AMS MSC: | 26A06 (Real functions :: Functions of one variable :: One-variable calculus) |
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Pending Errata and Addenda
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