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

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?

• do you want the power series about $x=0$ or $x=1$? – J. W. Tanner Dec 1 '19 at 20:57

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 $$1$$.
• Don't $\sum \frac{x^n}{2^{n+1}}$ and $\sum (x-1)^n$ have intervals of convergence that overlap? I don't see how these two expressions can both approximate the original function well – user162520 Dec 1 '19 at 21:06
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