= Solution
A smooth $2\pi$-periodic function has coefficients $g_k=(2\pi)^{-1}\int_0^{2\pi}g(x)e^{-ikx}dx$ decaying faster than any power, by repeated integration by parts. Thus its <Fourier series> and every differentiated series converge absolutely and uniformly to the corresponding function and derivatives. Solving each spatial <Fourier mode> gives
$$
\boxed{u(t,x)=tg_0+\sum_{k\ne0}g_k\frac{\sin(kt)}k e^{ikx}}.
$$
The zero-frequency term is linear in time, not obtained by division by zero. Uniform convergence on compact time intervals permits termwise differentiation and verifies the initial conditions and the <wave equation>.
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