Taylor series of a power series.

Question

Consider a power series $f(x)$ around a point $c \neq 0$. Then is it equal to its Taylor series around $0$? The reason I am wondering about this is because if it is true even for some special cases, then it means there is a power series around $c \neq 0$ which is also a power series around $0$. Then if $f(a)$ converges for some $a$, it implies $f$ converges absolutely and uniformly on $(-a, a)$ and $(c - a, c + a)$ which is counter intuitive.

Answer 1

If by $\sum a_nx^n=\sum b_n(x-c)^n$ you mean rhat the series converge to the same function $f(x)$, then we can take for example $f(x)=x^{17}$, or $f(x)=e^x$, and $c$ any non-zero number.