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Counter example: probability distribution of $X_n$ given by $$P_n(x)=\frac{3n}{(1+nx)^4},\;\;x\geq 0,$$ properly normalized to unity. Then $\mathbb{E}[X_n]=1/2n$ and $\mathbb{E}[X_n^2]=1/n^2$ both vanish in the limit $n\rightarrow\infty$, but higher moments and cumulants diverge.