I'm trying to using Kolmogorov's 3 series to show that if

$$ X_i \quad iid \sim N(0, \frac{1}{\sqrt{i}} ) $$

Does

$$ \sum_i^\infty X_i \quad \text{converge?} $$

Given that the sum of the variances $\sum_1^\infty \frac{1}{i^\frac{1}{2}}$ diverges, I'm thinking no it doesn't converge.

Is there a way to show this using Kolmogorov's 3 series?