Are we guaranteed that the harmonic series minus infinite random terms always converge? Consider the known harmonic series
$\sum_{n=1}^\infty \frac{1}{n}$ and modify it as follows
$$\sum_{n=1}^\infty a_n\frac{1}{n}$$
where
$$a_n \sim \operatorname{Bern} \left({\frac{1}{2}}\right)$$
i.e. each $a_n$ is $0$ or $1$ with probability $\frac{1}{2}$ (basically what one is doing here is removing randomly an infinite number of terms of this series)
Is it possible to know that this series is almost always convergent? Meaning that it converges with probability $1$?

For any such sequence $(a_n)$, the corresponding sequence $(b_n)$ given by $b_n=1-a_n$ is just as likely. So according to your intuition, the series
$$\sum_{n=1}^\infty b_n\frac{1}{n}$$
should converge almost surely. But then
$$\sum_{n=1}^\infty \frac{1}{n} = \sum_{n=1}^\infty a_n\frac{1}{n} + \sum_{n=1}^\infty b_n\frac{1}{n}$$
would converge too!
