Does interval of convergence of a power series always contain the origin?

Question

In Mathematical Methods in the Physical Sciences by Mary L. Boas, it was stated that

Since the interval of convergence of a power series contains the origin, $\sin x= a_0+a_1x+a_2x^2+... $ must hold when $x=0$.

The author is trying to demonstrate how to find a power series for a given function.

But this seems to suggest if a power series has an interval of convergence, it will contain the origin.

However according to this post, not all power series has an interval of convergence that contain the origin. Is the author being sloppy here?

Answer 1

If the power series has the form $\sum_{n=0}^{\infty}a_nx^n$, then the intervall of convergence always contains $0$.

If the power series has the form $\sum_{n=0}^{\infty}a_n(x-x_0)^n$, then the intervall of convergence always contains $x_0$.

Answer 2

A power series $\sum a_n x^{n}$ can have radius of convergence $0$, but it always converges at $0$. When $x=0$ the series is simply $a_0+0+0+0+\cdots$ which is convergent.