= Solution
Write $r$ for a positive <integer> moment index. The <moment> is
$$
\mathbb E[X^r]=\theta c^\theta\int_c^\infty x^{r-\theta-1}\,dx.
$$
This <improper integral> converges exactly when $r-\theta-1<-1$, namely $r<\theta$. For those indices,
$$
\boxed{\mathbb E[X^r]=\frac{\theta c^r}{\theta-r},\qquad r\in\mathbb Z_{>0},\ r<\theta.}
$$
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 $\theta\leq1$, no positive integer <moment> is finite.
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