The Mysteries of the Infinite

We’re about to start thinking about the final topic of math the: the mysteries of infinity. Can we, despite our own finitude, understand something about the infinite? Can we enter into the swirling mysteries of the infinite, and rather than being pulled under and drowned in its vortex, actually emerge victorious? What, after all, is:

To start, I want you to spend the weekend thinking about these three infinitely-long sums (or series, to use the fancy math word):

\[\begin{align*} &1 + \frac{1}{2} + \frac13 + \frac14 + \frac15 + \frac16 + \cdots \\ \\ &1 + \frac12 + \frac14 + \frac18 + \frac{1}{16} + \frac{1}{32} + \cdots \\ \\ &1 + \frac{1}{4} + \frac{1}{9} + \frac{1}{16} + \frac{1}{25} + \frac{1}{36} + \cdots \end{align*}\]

What happens to them? There are an infinite number of things added up, so are these all infinite? Or are any of them finite? Why? Can you figure out?!?

You might have seen some of these before; if so, try not to remind yourself of your previous knowledge, and definitely don’t look anything up. Come up with ideas, try things, test things; see how far you can go trying to wrangle and rassel these creatures by yourselves!