Suppose $X_n$ are independent Cauchy r.v.s. I'm trying to prove that $\limsup \log X_n/\log n = c$ almost surely for some constant $c$. I know that by Borel-Cantelli it suffices is prove that
$\sum\mathbb{P}(X_n\ge n^c)=\infty$ and $\sum\mathbb{P}(X_n\geq n^{c+\epsilon})<\infty$ $\forall\epsilon>0$
However $\mathbb{P}(X_n\geq n^c)=\frac{1}{2}-\frac{1}{\pi}\arctan(n^c)$ and I know no way of evaluating this sum. Are there any convergence tests I should use that I've managed to miss? Or have I made a silly error somewhere? Thanks!