On Wikipedia, an example is given where the weak law of large numbers holds but the strong law does not. If $X_n$ are i.i.d. exponential random variables with parameter 1 and $Y_n = \frac{\sin(X_n)}{X_n}e^{X_{n}}$, then
$$\frac{\sum_1^nY_{k} - 2/\pi}{n}\rightarrow 0 \hspace{.5cm}\text{(in probability)}$$
To see this, note that we have $y\mathbb{P}(|Y| > y) \le y\int_{\ln(y^{2})}^{\infty}e^{-t}dt = y\frac{1}{y^{2}} = y^{-1}\rightarrow 0$ as $y\rightarrow\infty$, using the inequality $\frac{\sin(y)}{y}e^{y}\ge y\implies e^{y}\ge y^{2}$. Also, $\mathbb{E}(Y1_{|Y|\le n})$ can be shown to converge to $2/\pi$.
But this convergence does not hold almost surely. How do we know that the (conclusion of the) strong law fails? In this case, I want to know why the above convergence is not almost sure.