Choose alpha=1/4 and add height-one triangular spikes centered at n+1/2 with half-width 1/(4n) to x^-2. The supports are disjoint and lie strictly between consecutive integers, so f(n)=1/n^2 and the sample series converges. Each spike has area 1/(4n), so the improper integral diverges by the harmonic series. The baseline makes the function strictly positive and contributes only a finite integral.
