what is an example of a sequence of random variables $X_n$ that converges to $X$ in the first moment but not almost surely?
I think it is not possible!!
because we need to have, the probability that $X_n$ and $X$ differ, really small or that $X_n$ and $X$ differ very little in order to get convergence in the first moment. We don't want them to converge almost surely.
We want that they don't converge a.s so we want the probability that they differ large. Well we can only do so if they differ by little.
I can't think of an example.
Best Answer
Take a sequence of independent random variables $X_n$ where $\mathbb P(X_n=1)=\frac1n$ and $\mathbb P(X_n=0)=1-\frac1n$:
Then $X_n$ converges to $0$ in first moment (I would say "in mean") since $\mathbb E[|X_n-0)|]= \frac1n \to 0$, and indeed $X_n$ converges to $0$ in all moments
But $X_n$ does not converge to $0$ almost surely, since for all $N$ and $\epsilon$ you have $\sum\limits_{n=N+1}^{\infty} P(|X_n-0| \ge \min(1,\epsilon))=\infty $ and so $\mathbb P\left( \lim\limits_{n\to\infty}\! X_n = 0 \right) = 0 \not = 1$