Solution
= Solution
{c}
For fixed $\varepsilon>0$ and $n>\varepsilon$,
$$
\mathbb P(|X_n-1|>\varepsilon)=\frac1{n\log n}\to0,
$$
so there is <convergence in probability> to one. The absolute difference is exactly $n$ on the exceptional event, hence
$$
\boxed{\mathbb E|X_n-1|^p=\frac{n^{p-1}}{\log n}.}
$$
For every $p>1$ this tends to infinity, so <convergence in Lp> fails. In contrast its first moment is $1/\log n\to0$, so convergence actually holds in $L^1$.