= Solution
The original PDF has height \b[$1/n$] on the interval $[0,e^n]$. For finite $p$,
$$
\|f_n\|_p=\frac{e^{n/p}}n\longrightarrow\infty,
$$
so the sequence cannot converge in $L^p$, since a norm-convergent sequence is bounded. For $p=\infty$, $\|f_n\|_\infty=1/n\to0$, so \b[it converges to zero in $L^\infty$]. It also converges pointwise, and hence <almost everywhere>, to zero. Finally for each $\epsilon>0$, once $n>1/\epsilon$ the set $\{|f_n|>\epsilon\}$ is empty. Thus \b[it converges in <measure> to zero], even on this infinite-measure space.
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