How do you find the power series representation for the function $f(x)=(1+x)/(1-x)$ ?

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Answer 1 · 2014-09-09T18:50:09

Recall:
$1/{1-x}=sum_{n=0}^infty x^n=1+sum_{n=1}^infty x^n$
By multiplying by $x$ ,
$x/{1-x}=sum_{n=0}^infty x^{n+1}=sum_{n=1}^infty x^n$

So,
${1+x}/{1-x}=1/{1-x}+x/{1-x}=1+sum_{n=1}^infty x^n+sum_{n=1}^infty x^n$
$=1+2sum_{n=1}^infty x^n$

How do you find the power series representation for the function f(x)=(1+x)/(1-x)f(x)=(1+x)/(1-x) ?