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My thinking is that all geometric series are power series, since the form of a term of a power series is $c_n(x-a)^n$. If you look at $c_n$ as the starting constant and $x$ as the common ratio (and $a$ being $0$), then it matches the formula for a term of a geometric series, $ar^{n-1}$.

But if $a$ is zero, that makes the power series centered around zero. Thus my conclusion that every geometric series is a power series centered at zero. Is this correct?

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Every geometric series given by $$a+ar+ar^2+\dots$$ is simply the power series representation of the function $$f(r)=\frac{a}{1-r}$$ centred at $r=0$.

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Yes, your thinking is correct here.

But note that the notion of the power series (even about a given point $0$) is much more powerful and general than the notion of the geometric series.

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