Question

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

Answer 1

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`