Solution
= Solution
The coefficient ratio is
$$
\frac{n+1+2^{-(n+1)}}{n+2^{-n}}\longrightarrow1.
$$
Thus the <ratio test> again gives <radius of convergence> $\boxed{R=1}$. On $|z|=1$, however, the terms have absolute value $n+2^{-n}$, which does not tend to zero. The necessary term condition for a <convergent series> fails at every boundary point. Therefore this <power series> \b[diverges everywhere on $|z|=1$].