It is known that the series \(\sum_{n=1}^\infty \frac{1}{n}\) diverges while the series \(\sum_{n=1}^\infty \frac{(-1)^n}{n}\) converges. One may ask about the convergence or divergence of the series \(\sum_{n=1}^\infty \frac{\xi_n}{n}\), where \(\xi_n\) are independent random variables taking the values \(+1\) or \(-1\) with the probabilities \(p\) and \(q = 1 - p\) correspondingly.
Show that if \(p = q = \frac{1}{2}\), the series converges with the unit probability.
Show that otherwise the series diverges with the unit probability.