Solution
= Solution
For $\alpha\ne-1$,
$$
\int_1^Rx^\alpha\,dx=\frac{R^{\alpha+1}-1}{\alpha+1}.
$$
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
$$
\boxed{\alpha<-1}.
$$