Use Zeno's paradox to motivate the discussion of the sum of the geometric series -- What is wrong with Zeno's proof that the turtle will overcome Achilles?
So the sum of the geometric series 1 + 1/2 + 1/4 + ... converges? How do you know that?
If someone says that it is because the numbers in the series get smaller and smaller, that is great. If not, give that as a claim that someone made and ask if people agree.
Then show the harmonic series and ask people if it converges. Use some computation and perhaps a computer to have the group start believing that it doesn't. Then ask how you could prove this.