<s>[INST] Which expression about power series is correct? I found something weird about power series.
In my textbook,
$$ \frac{1}{1-x} = 1+x+x^2+x^3+\cdots=\sum_{n=0}^\infty x^n \quad |x|<1  $$
and $ \frac{1}{2+x} $ could be expressed using above equation.
$$ \frac{1}{2+x} = \frac{1}{2 \left(1 + \frac{x}{2} \right)} = \frac{1}{2 \left[ 1- \left( - \frac{x}{2} \right) \right]} \\ = \frac{1}{2} \sum_{n=0}^\infty \left( - \frac{x}{2} \right)^n = \sum_{n=0}^\infty \frac{(-1)^n}{2^{n+1}}x^n $$
which has interval of convergence $ (-2,2) $.
And below is my solution.
$$ \frac{1}{2+x} = \frac{1}{1-(-1-x)} = 1+(-1-x)+(-1-x)^2+(-1-x)^3+\cdots \\ =\sum_{n=0}^\infty (-1-x)^n \quad |-1-x|<1 $$
which has interval of convergence $ (-2,0) $
Here are my questions.


*

*Why above two power series have different interval of convergence from one equation $ \frac{1}{2+x} $? Why does it appear?

*How do I find out right answer?

 [/INST] You are just comparing two series developed around different points:
 - one around $x=0$;
 - the other around $x=-1$.
and remember that the radius of convergence is the distance from that point to the next irregularity ($x=-2$ in this case).
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