I want to determine for which $p\geq 1$ and $f_n=n \cdot 1_{[n,n+1/n^2]}$ we have $$\lim_{n\rightarrow\infty} ||f_n||_p=0$$
I have gotten to: $$||f_n||^2=\bigg( \int |n\cdot 1_{[n,n+1/n^2]}|^p \bigg)^{1/p}$$ $$=\bigg( \int n^p \cdot 1_{[n,n+1/n^2]} \bigg)^{1/p}$$ $$=\bigg( n^p \cdot 1/n^2 \bigg)^{1/p}$$ $$=\bigg( n^p/n^2 \bigg)^{1/p}$$ $$= n/n^{2/p} $$ $$= 1/n^{2/p-1} $$
So only for p=1 it seems that this is true.
However this seems as a strange result as the integral should converge
Any hint would be appreciated