= Solution
Choose $0<\rho<r$ and traverse the <circle> $|z|=\rho$ counterclockwise, so it has <positive orientation of a contour>. The <Cauchy coefficient formula> gives, for every <nonnegative integer> $n$,
$$
\boxed{c_n=\frac{1}{2\pi i}\int_{|z|=\rho}\frac{f(z)}{z^{n+1}}\,dz,\qquad 0<\rho<r.}
$$
The smaller <radius> matters: the <holomorphic function> is only known on $|z|<r$, so its values on $|z|=r$ need not exist. To see directly why this <contour integral> selects $c_n$, the <power series> converges uniformly on $|z|=\rho$, allowing <uniform convergence and contour integration> to justify integrating term by term. The <contour integral> of $z^{k-n-1}$ is zero for $k\ne n$ and $2\pi i$ for $k=n$. Thus all other terms disappear.
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