CK12 Calculus - Single Variable

2. Find the Maclaurinseriesof for allx.

Hint: wouldyou dividexbyex?

3. Find the Maclaurinseriesofcos 3xandsin 3xfor allxusingEuler'sformula.


4. Find expressionsfor the series and for allθand |x| < 1 usingEuler's
   formula.

BinomialSeries
We havelearnedthe BinomialTheoremfor positiveintegerexponents:

wherethe Binomialcoefficientsare denotedby

and for

AS a simpleBinomialfunction,takea= 1 andb= x. Then

Letrbe a real numberand f(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

wherethe Binomialcoefficientsare denotedby

and for
As a simpleBinomialfunction,takea= 1 andb=x. Then

.