= Solution
{c}
Take $0\leq r<R$. At the point $r$, the <power series> has <absolute convergence>, so $\sum_{n=0}^{\infty}|a_n|r^n<\infty$. On the whole <interval> $[-r,r]$,
$$
|a_nx^n|\leq|a_n|r^n.
$$
The <Weierstrass M-test> with $M_n=|a_n|r^n$ proves \b[<uniform convergence> on $[-r,r]$.] It also gives the explicit tail estimate
$$
\boxed{\sup_{|x|\leq r}\left|\sum_{n=N+1}^{\infty}a_nx^n\right|\leq\sum_{n=N+1}^{\infty}|a_n|r^n\longrightarrow0.}
$$
For $r=0$ the <interval> consists only of zero, and all positive-degree terms vanish. If $R=\infty$, the same argument applies to every finite $r$; if $R=0$, there is no $r\geq0$ with $r<R$.
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