$E(X_n)\rightarrow E(X)\nRightarrow E(|X_n-X|)\rightarrow 0$ If $\{X_n\}$ is a sequence of random variables and $X$ be another random variable, then as far as I know $$E(X_n)\rightarrow E(X)\nRightarrow E(|X_n-X|)\rightarrow 0$$ always. 
But I am not getting an example. Can somone provide me one?

Well, just pick two different measurable functions $Y$ and $X$ with the same integral and set $X_n=Y$. Then $E[X_n]\equiv E[Y]=E[X]$ but $E[\lvert X_n-X\rvert]\equiv E[\lvert Y-X\rvert]>0$.
