Boundary Values & Fourier Series · Hard problem

A rod $0 \le x \le \pi$ obeys $u_t = u_{xx}$ with $u(0, t) = u(\pi, t) = 0$ and uniform initial temperature $u(x, 0) = 1$. Find $\displaystyle\lim_{t\to\infty}e^{t}\,u\!\left(\dfrac\pi2, t\right)$.