Changes

::<math>u(x,t) = \sum_{n = 1}^\infty a_n(t) \sin(nx) + \sum_{n=0}^\infty b_n(t) \cos(nx)</math>.

::<math>u(x,t) = \sum_{n = 1}^\infty a_n(t) \sin(nx) + \sum_{n=0}^\infty b_n(t) \cos(nx)</math>.

This is the so-called Fourier expansion, which plays a foundational role in modern math and physics.  Once the initial condition is expressed in this form, Fourier concluded that it was easy to solve for the functions <math>a_n(t)</math> and <math>b_n(t)</math> and thus solve the heat equation.  Today it is understood that Fourier's work was essentially correct, though deficient in certain technical details which could not be fixed until the advent of modern techniques of [[analysis]].

This is the so-called Fourier expansion, which plays a foundational role in modern math and physics.  Once the initial condition is expressed in this form, Fourier concluded that it was easy to solve for the functions <math>a_n(t)</math> and <math>b_n(t)</math> and thus solve the heat equation.  Today it is understood that Fourier's work was essentially correct, though deficient in certain technical details which could not be fixed until the advent of modern techniques of [[analysis]].