<s>[INST] Two different series for $\frac{1}{x + 2}$ 
Find the appropriate power series for $f(x) = \frac{1}{x + 2} $

First method :  $\frac{1}{x + 2} =  \frac{1}{2(1 - (-\frac{x}{2}))} = \sum_{n=0}^\infty \frac{(-1)^n}{2^{n+1}}x^n$ , $|x|\lt2$
Second method : $\frac{1}{x + 2} = \frac{1}{1 - (-x-1)} = \sum_{n=0}^\infty (-x-1)^n$ , $-2 \lt x \lt 0$
Is there any problem in my answers ? Why these power series have different radii of convergence ?
 [/INST] The first one is the series centered at $x=0$ while the second one is the series centered at $x=-1$ therefore the radius of convergence are different.
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