Almost alternating harmonic series converges or not almost surely?

Question

Consider $\sum_{i} I_i/i$, where $I_i$ is defined by a coin toss to get $-1$ and 1 values for $I_i$, heads meaning 1 and tails meaning $-1$. Is $\sum I_i/i$ converging almost surely?

Answer 1

There are at least two approaches:

Approach 1: Apply Kolmogorov's convergence theorem.

Approach 2: Show that $$M_n := \sum_{j=1}^n \frac{I_j}{j}$$ defines an $L^2$-bounded martingale. Conclude that the limit $\lim_{n \to \infty} M_n$ exists almost surely.

Remark: Although not explicitely mentioned in your question, I take it that the coin is fair (i.e. $\mathbb{P}(I_j=1)=\mathbb{P}(I_j=-1)=1/2$) and that the random variables $I_j$, $j \geq 1$, are independent. Otherwise the random series will clearly, in general, fail to converge.