Suppose that $X_k \in \Gamma(3,k)$, that is, that $$ f_{X_k}(x) = \begin{cases} \frac{x^2}{2k^3} e^{-\frac{x}{k}} \quad & \text{for } x > 0, \\ 0 & \text{otherwise.} \end{cases}$$ (I suppose these are the densities.) Show that $$\sum_{k=1}^n \frac{1}{X_k} - \frac12 \log{n}$$ converges in probability as $n \to \infty$.
This is a problem from Allan Gut's book. It's is hinted that one should remember $$\lim_{n \to \infty} \left(\sum_{k=1}^n \frac{1}{k} - \log{n} \right) = \gamma$$ but I don't quite see how that comes into play. My main problem is that I don't see a way to guess a limit and can't see how to prove that it's Cauchy (in Probability) either. Any solutions/hints would be nice.