Given $k>0$ and $u_0\in\mathcal{C}^1[0,L]$, $u_0\geq 0$ we have the problem $$\left\{\begin{matrix} \frac{\partial u}{\partial t}(x,t)=k\frac{\partial^2u}{\partial x^2},&0<x<L,&t>0\\ \frac{\partial u}{\partial x}(0,t)=\frac{\partial u}{\partial x}(L,t)=0,&t>0\\ u(x,0)=u_0(x),&0\leq x\leq L \end{matrix}\right.$$
Spliting variables I get the solution $$u(x,t)=\sum_{n=0}^\infty a_n\cos{(\frac{n\pi x}{L})}e^{-(\frac{n\pi}{L})^2kt}$$ where $$a_n=\frac{2}{L}\int_0^Lu_0(s)\cos(\frac{n\pi s}{L})ds$$ after that I want to show that this series converges and also that this is actually a classical solution of the problem, there is where I am stuck.