Past exam of the year 1 of the computer science course of the University of Oxford / 2019 / Discrete mathematics, continuous mathematics, and probability resit / 7 / a / iii

Write $r$ for a positive integer moment index. The moment isThis improper integral converges exactly when $r-\theta -1<-1$, namely $r<\theta$. For those indices,At $r=\theta$ the divergence is logarithmic; for $r>\theta$ it is a power divergence. Thus, in the source's moment notation, the positive integers $k$ allowed are exactly $k<\theta$. If $\theta \le 1$, no positive integer moment is finite.