Example of a function

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

Solution

Define $f$ by

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

Hence

$$\int_{1}^{\infty} f(x) \, \mathrm{d}x = \int_{1}^{\infty} \mathrm{e}^{-x} \, \mathrm{d}x + \sum_{n=2}^{\infty} \frac{1}{n^2}$$

which clearly converges , but on the other hand

$$ \int_{1}^{\infty} f^2(x) \, \mathrm{d}x \geq \sum_{n=2}^{\infty} \int_1^{\infty} \max\{0, (n - n^4|x - n|)^2 \} \, \mathrm{d}x = \sum_{n=2}^{\infty} \frac{2}{3n} = \infty$$

The basic idea is to construct 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 $\sim n^{-1}$.