I understand that any power series should converge at one point, namely the center(?) of the series. For instance, $x=a$ in the series $\sum_{n=0}^\infty c_n(x-a)^n$
But is there a power series that converges at exactly one point? I can't think of any examples myself but think that somehow if we applied ratio test and got radius of convergence 0 we might be able to.
So$$lim_{n\rightarrow\infty}|\frac{c_{n+1}(x-a)^{n+1}}{c_n(x-a)^n}|=lim_{n\rightarrow\infty}|\frac{c_{n+1}(x-a)}{c_n}|\ge1$$ for any $x\ne a$ for some $c_n$ and $a$, but what could $c_n$ possibly be? Maybe $c_n=n!$ but I'm not sure if that's even a valid series.