Past exam of the options II course of the University of Oxford / 2009 / D / D3 / i / b / Solution

For $\alpha \ne -1$,The limit as $R\to \infty$ is finite exactly when $\alpha <-1$; the case $\alpha =-1$ diverges logarithmically. Positivity and the monotone convergence theorem identify these limits with the Lebesgue integral. Hence