Many thanks. Very nice answer. Before accepting, can we say something about the existence of moments for $p<0$ using the same argument?

In addition, by wildly interchanging the series and the integration process, I came up to the conclusion that all the moments exist. But I would like to confirm this.

@Davide Thanks. I know that I could try by integrating the density or by using $E[X^n]=\int n x^{n-1}P(X>x)$. However, since both approaches involve the integration of an alternating series, I was looking for a credible reference that justifies the steps. My google-foo has not lead me to a helpful reference yet.