We know that:
- $\sum \limits _{n=1}^{\infty}\frac{1}{n} = \infty$
- $\sum \limits _{n=1}^{\infty}\frac{(-1)^n}{n} < \infty$
Both are easy to show. In the first case we can use the criterion based on integrating the function $f(x) = \frac{1}{x}$ and the convergence of the second series can be settle with the Dirichlet's criterion.
Let's imagine a random series:
$$\sum \limits _{n=1}^{\infty}\frac{\lambda_n}{n},$$
where $\lambda = 1$ or $\lambda = -1$.
$P(\lambda = 1) = P(\lambda = -1) = \frac{1}{2}$.
When the series will converge?
If the random variables $\lambda_n$ are independent and take the values $1$ and $-1$ with probability $1/2$ each then $\sum \lambda_na_n$ converges almost surely if and only if $\sum a_n^{2}$ is finite. In particular this holds for $a_n=1/n$. However the series may not converge almost surely if independence is dropped: if all the $\lambda_n$ are equal then the series diverges almost surely.