wherethe Binomialcoefficientsare denotedby
and for
As a simpleBinomialfunction,takea= 1 andb=x. Then
.
Letrbe a real numberandf(x) = (1 +x)r. Isf(x) equalto a seriesin the form of (BE) exceptthat theremay
be an infiniteseries?The answeris yes.
Theorem(BinomialSeries)Letrbe a real numberand |x| < 1. Then
where and for
Example 1 Find a powerseriesrepresentationof.
Solution.We needto computethe Binomialcoefficientsfor
So if
Example 2 Find a powerseriesrepresentationof wheremis a positiveinteger.
Solution.We needto computethe Binomialcoefficientsforr= -m(and will replacexby -x)