Rookie question - Series 1/n

Hi there,

I've read through why the sum of the series 1/n ( n=1 to infinity) diverges. Ok but still it is mindboggling because the term of the series will get smaller and smaller surely never being able to reach infinity, yet it is clear that it TENDs to infinity.. just very wierd because I would have thought there could be a point where it could get so tiny that it would maybe converge to some value?

Just interested in hearing a different take on this series compared to my text book and if there is some example in the real world where this series is applied

Thank you