Consider the heat equation
$$\frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial x^2}$$
for $x \in \mathbb{T}:=\mathbb{R}/2\pi\mathbb{Z}, t \geq 0$, with initial condition $u(x,0)=f(x)$ for some $f \in L^1(\mathbb{T})$.
I'm asked to verify that $\sum_{n \in \mathbb{Z}} \hat{f}(n) e^{-n^2 t} e^{inx}$ satisfies this equation for all $x \in \mathbb{T}, t>0$ (where the hat denotes Fourier coefficients).
I'm not sure how to justify differentiating term-by-term with respect to $t$, because I don't know how to prove (or if) the series $\sum_{n\in\mathbb{Z}} n^2 e^{-n^2 t}$ converges uniformly for $t>0$ so that I can apply the Weierstrass M-Test (the Fourier coefficients are bounded because they tend to zero). Perhaps the M-Test is not sufficient. I'd appreciate it if someone could tell me how to justify term-by-term differentiation here.
Certainly not uniformly on $(0,\infty)$, but uniformly on $[\varepsilon, \infty)$ for each $\varepsilon > 0$ should be enough.