How can I prove the convexity of the function in $(0,\infty)$ for $a,b,\sigma >0$ ?$$f(x)=(a+bx)\sum_{k=0}^{\infty}e^{-k^2x^2/\sigma^2}$$
PS: 1. The function is actually convex but only if there are infinitely many squared exponentials. Hence the proof must be using this fact somewhere. 2. Limiting value of the function and its first two derivatives can be proven to be $\infty,-\infty,\infty$ respectively. 3. Final aim is to minimize the function. So if not convexity, proof of its decreasing nature from $x=0^+$ to the minima will suffice. (That is, convexity, quasi-convexity or even a proof of single minima would suffice)