finding series representation with constant in numerator

Question

how is the method for finding series rep when you have constant in numerator (and variable)

couldnt find anything when searching or in my mathbook

question: express (3+x)/(1+x) in terms of (x-1)^k

i know that you are supposed to get the expression in a form of a/(1-r), but now i have "r" also in numerator

Help is very appreciated!

Answer 1

$$\frac{x+3}{x+1}=1+\frac{2}{x+1}=1+\frac{2}{2+x-1}=1+\frac{1}{1+\frac{(x-1)}{2}}$$

Then by the geometric series you have

$$\frac{x+3}{x+1}=1+\sum_{n=0}^{\infty}(-1)^n(\frac{x-1}{2})^{n}$$

which converges when $|x-1|<2$.

Answer 2

Hint:

$$\frac{3+x}{1+x}=1+\frac2{2+(x-1)}.$$