I can't solve this exercise:
Consider the sequence of function $f_n:[1,+\infty\rangle > \to\mathbb{R},\;n\in\mathbb{N}$
$$f_n(x):=\frac{x-1}{\left(x+\frac{1}{n}\right)^{3/2}\ln(x+\frac{1}{n})}$$
Study the convergence properties of the sequence (pointwise almost everywhere, srong in $L^p(1,+\infty)$ for $1\le p\le +\infty$, weak for $1\le p< +\infty$, $\mathrm{weak}^{*}$ in $L^{\infty}$)
For the pointwise limit a.e. I obtain:
$$f_n(x)\rightarrow f(x)=\frac{x-1}{x^{3/2}\ln(x)}\quad\forall x\in (1,+\infty)$$
But as soon as I try to check if there is some $1\le p < +\infty$ such that
$$\lim_{n\to +\infty}||f-f_n||_{L^p}=0$$
i.e.:
$$\lim_{n\to +\infty}\int_1^{+\infty}|f(x)-f_n(x)|^p dx=0$$
I don't know how to go on. Same problem with $p=+\infty$.
