= Solution
For the <Hamiltonian> $p^2/(2m)+V(q)$, the real-time <configuration-space path integral> is
$$
\langle q_f|e^{-iHt/\hbar}|q_i\rangle=\int_{q(0)=q_i}^{q(t)=q_f}\mathcal Dq\,\exp\left[\frac i\hbar\int_0^t\left(\frac m2\dot q^2-V(q)\right)dt'\right],
$$
with its measure defined by time slicing. <Wick rotation> gives the <Euclidean path integral> and the thermal trace.
For the <infinite square well> on $(0,L)$, the <Dirichlet boundary conditions> select the normalized <energy eigenstates>
$$
\psi_n(q)=\sqrt{\frac2L}\sin\frac{n\pi q}{L},\qquad E_n=\frac{\hbar^2\pi^2n^2}{2mL^2},\qquad n=1,2,\ldots.
$$
There is no $n=0$ state: the corresponding sine is identically zero. Taking the <trace> in this <orthonormal basis> gives the <canonical partition function>
$$
\boxed{Z(\beta)=\sum_{n=1}^\infty\exp\left(-\frac{\beta\hbar^2\pi^2n^2}{2mL^2}\right)}.
$$
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