Show that (9) in Sec. 12.6 with coefficients (10) is a solution of the heat equation for t > 0 assuming that f(x) is continuous on the interval 0 ≤ x ≤ L and has one-sided derivatives at all interior points of that interval. Proceed as follows.

Show that |B_n| is bounded, say |B_n|

and, by the Weierstrass test, the series (9) converges uniformly with respect to x and t for t ≥ t_0, 0 ≤ x ≤ L. Using Theorem 2, show that u(x, t) is continuous for t ≥ t₀ and thus satisfies the boundary conditions (2) for t ≥ t₀.