Is the sequence $$\Bigl( \frac{1}{n+1} \Bigr)^\alpha,\quad n \geq 0$$ a moment sequence for any $\alpha \in (0,1]$ for some random variable on [0,1]?
We have tried checking whether the corresponding matrix is positive definite, this did not show that it was not a moment sequence.
HINT:
Yes, it is, see Hausdorff moment problem. Also, this follows from the fact that the function
$$ s \mapsto \frac{1}{(1+s)^{\alpha}}$$
is totally monotone ( Bernstein theorem).
In fact you can produce a density on $[0,1]$ with its moment sequence the given one. You only need to find the inverse Laplace transform of $\frac{1}{(1 + s)^{\alpha}}$.