Zeno's Paradox and Infinite Sums

Can you cross a room if you must first travel half the distance, then half again, forever? Zeno of Elea argued all motion is impossible, and his ancient paradox opens the door to profound questions about infinity. In this lecture, Joel David Hamkins traces a path from Zeno's challenge through the mathematics of infinite sums. He asks whether 0.999... truly equals 1, unpacks what decimal notation actually means, and derives the geometric series formula to resolve Zeno's puzzle. But not all infinite sums behave so neatly. The harmonic series diverges to infinity, and its alternating cousin converges, yet Riemann's rearrangement theorem reveals something fascinating: when you add infinitely many numbers, the order in which you add them can change the result entirely. This lecture sets the stage for a deeper exploration of supertasks and the strange logic of the infinite.