# Is it correct to say that all geometric series are power series centered at $0$?

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?

## 2 Answers

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 $$r=0$$.

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.