Suppose we have a sequence $(X_n)_{n \in \Bbb N^*}$ that follows the distribution $\frac{1}{n}\delta_{n^2}+(1-\frac{1}{n})\delta_0$.
What's the expected $E(X_n^p)$ with $p \geq 1$???
My attempt : $$E(X_n^p)=\sum_{n \in \Bbb N^*} n^p P(X_n=n) $$ which is equal to : $$E(X_n^p)=\sum_{n \in \Bbb N^*} \frac{n^p}{n^2}=\sum_{n \in \Bbb N^*} n^{p-2} $$ which goes to infinity.