Different ways to find power series for $\frac{1}{2-x}$

Question

I am trying to find the power series representation for $\frac{1}{2-x}$.

I realize that I can write it as $\frac{1}{2}\cdot \frac{1}{1-\frac{x}{2}}$ and then use the geometric series.

My question is, what is wrong with writing it as $\frac{1}{1-(x-1)}$ and then saying the power series is $\sum(x-1)^n$?

Does the term $t$ in the expression $\frac{1}{1-t}$ always need to be a monomial to apply this trick? Why?

Answer 1

It all depends on where you want to center your power series.

The power series $\sum(x-1)^n$ is centered in $x_0=1$, meaning its truncates give good results (i.e. approximate well your function) near $1$.

Answer 2

As pointed out, these two power series are both valid, just centered at different points. It should be noted that $\sum (x-1)^n$ has a radius of convergence of $1$, so this is valid on $(0,2)$ while $\frac{1}{2}\sum (x/2)^n$ has a radius of convergence of 2, so this is valid on $(-2,2)$. The two power series agree where they are both defined