Consider the Theta function
$\theta(x,t)=\sum\limits_{m=-\infty}^{\infty}K(x+2m,t)$,
where $K(x,t)=\frac{e^{-\frac{x^2}{4t}}}{\sqrt{4 \pi t}}$. The $\theta$ function is seen in the solution of the heat equation
$u_t=u_{xx}$, $0<x<1$, $t>0$
$u(x,0)=f(x)$, $0<x<1$,
$u(0,t)=u(1,t)=0$, $t>0$ as
$u(x,t)=\int\limits_0^1[\theta(x-\xi,t)-\theta(x+\xi,t)]f(\xi)d\xi$.
I want to show that the series for the function $\theta$ is uniformly absolutely convergent.
My Work: By using the inequality $\exp(-x)\leq p!x^{-p}$, $p=1,2,\ldots$, $x>0$, for $p=2$, I have
$|\theta(x,t)|\leq\frac{16t^{\frac{3}{2}}}{\sqrt{\pi}}\sum\limits_{m=-\infty}^{\infty}\frac{1}{(x+2m)^4}$
At this point I am confused, the majorizing series is a $p$-series but the variables $x$ and $t$ confuses my mind.
Question: Could any one please help me to show uniform absolute convergence of the series for $\theta$ and it's partial derivatives.
Thanks in advance!..