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Show that |∂un/∂t| t0 and the series of the expressions on the right converges, by the ratio test, Conclude from this, the Weierstrass test, and Theorem 4 that the series (9) can be differentiated tem by term with respect to t and the resulting series has the sum ∂u/∂t. Show that (9) can be differentiated twice with respect to x and the resulting series has the sum ∂2u/∂x2. Conclude from this and the result to Prob. 19 that (9) is a solution of the heat equation for all t > t0. (The proof that (9) satisfies the given initial condition can be found in Ref. [C10] listed in App.1.)

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