This question already has an answer here:
The proof involving partial sums up to the nth term, where n is some power of $2$, completely makes sense. But just looking at the series itself, it seems very strange that it's divergent.
For large values of $n$, $a_n$ would start being extremely small and having an indistinguishable effect on the overall sum. All the sixth sense I've gained from working with limits makes it seem really strange that this would be considered divergent.
Surely there is a number (not even that difficult to find) such that we don't have enough computational power to calculate it's difference with the next terms (seeing as we'd be calculating differences based on hundreds of decimal places).
If you've any intuition on this I'd very much love to hear it!
Edit: I'm not asking for the proof of why it's divergent, I'm asking for peoples' personal ways of thinking and making sense of this intuitively. The post suggested to have been duplicated presents formal proofs; that's not what I'm looking for :)