I've been reading Chapter 24, "Uniform Convergence and Power Series" of Spivak's Calculus, and towards the end of the chapter there is the following theorem which I have a few questions about.
Theorem 6 Suppose that the series
$$f(x_0)=\sum\limits_{n=0}^\infty a_nx_0^n\tag{1}$$
converges, and let $a$ be any number with $0<a<|x_0|$. Then on $[-a,a]$, the series
$$f(x)=\sum\limits_{n=0}^\infty a_nx^n\tag{2}$$
converges uniformly (and absolutely). Moreover, the same is true for the series
$$g(x)=\sum\limits_{n\to\infty} na_nx^{n-1}$$
Finally, $f$ is differentiable and
$$f'(x)=\sum\limits_{n\to\infty} na_nx^{n-1}$$
for all $x$ with $|x|<|x_0|$.
Let me go through this result in my own words.
The series in (1) is an infinite series of numbers. The sequence being summed is $\{a_nx_0^n\}$, and it is summable by assumption. Do we call a power series evaluated at a certain $x_0$ a power series as well? Is the expression in (1) a power series?
Then we prove that on $[-a,a]$ the power series $(2)$ converges absolutely (when we evaluate it at any $x\in [-a,a]$), and also uniformly when we view it as a sum of functions. Is this correct?
Why did this theorem not just state that $f(x)$ converges absolutely for any $x$ in $(-|x_0|,|x_0|)$ and converges uniformly on this interval?