An interesting function

Give an example of a continuous function $f:[1, +\infty) \rightarrow \mathbb{R}$ such that $f(x)>0$ forall $x \in [1, +\infty)$ such that $\bigintsss_1^\infty f(x) \, {\rm d}x$ converges whereas $\bigintsss_1^\infty f^2(x) \, {\rm d}x$ diverges.

Solution

Define

$f(x) = e^{-x} + \sum_{n=2}^{\infty} \max\{0, n - n^4|x - n|\}$

It is easy to check that the given function has the properties in question. The idea behind that is to generate a train of picks. Picks are constructed to satisfy the following heuristics: Each pick at $x = n$ is of height $\sim n$ and width $\sim n^{-3}$ , thus the area is of order $\sim n^{-2}$ , when squared, however, its height gets squared and the area becomes of order $\sim n^{-1}$ .