= Solution
Put $x=\beta\epsilon$. The one-particle <canonical partition function> is
$$
z=e^x+1+e^{-x}=1+2\cosh x,
$$
and independence gives $Z=z^N$. Therefore the <independent symmetric three-level system> has
$$
\boxed{E=-\frac{\partial\log Z}{\partial\beta}
=-\frac{2N\epsilon\sinh x}{1+2\cosh x}}
$$
and, using part (c),
$$
\boxed{
S=Nk_B\left[
\log(1+2\cosh x)
-\frac{2x\sinh x}{1+2\cosh x}
\right]}.
$$
As $T\to\infty$, $x\to0$, all three levels become equally likely and
$$
E\to0,\qquad S\to Nk_B\log3.
$$
As $T\to0^+$, every particle occupies the nondegenerate ground level $-\epsilon$, and
$$
E\to-N\epsilon,\qquad S\to0.
$$
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