" Today I said to the calculus students, "I know, you're looking at this series and you don't see what I'm warning you about. you look at it and you think, 'I trust this series. I would take candy from this series. I would get in a car with this series.' But I'm going to warn you, this series is out to get you. Always remember: The harmonic series diverges. Never forget it.
For more about the harmonic series, here and here and one more here, but surely there are more...use the search function.