Confused about terminology for the center of power series; what does "centered at $x=1$" mean?

Question

I have just learned about power series and have mostly seen the center referred to as the $a$ value in the series of the form $\sum _{n=0}^\infty (x-a)^n$. However, I have also seen the $x$-value referred to as the center, e.g. in "compute the Taylor series of the function around $x = 1$". Does this last statement mean that the $a$ value in the Taylor series will be $1$? I am unsure because they are talking about $x$, not $a$; but $x$ seems to be referred to as the center but I have been taught $a$ is the center when we have a power series of this form.

Answer 1

The center is where the point where you evaluate your function is "near". Developping an analytic function near $x = 1$ means that you write it as, $$ f(x) = \sum_{n \geqslant 0} \alpha_n(x - 1)^n, $$ where $x$ is close to $1$ or equivalently, $$ f(1 + x) = \sum_{n \geqslant 0} \alpha_nx^n, $$ where $x$ is close to $0$ this time. Same thing with Taylor series development where you replace the infinite sum by a finite sum plus a little/great O. Therefore, with your notations, the center indeed refers to the $a$ but $x$ must be close enough to $a$ for the series to converge (at least when the radius of convegrence is finite).