Philosophy of Mathspeaking topic
Why does Grandi's series (1-1+1-1+...) seem equal to 0, to 1, and to 1/2 all at once, and what does it mean to 'assign a value' to a divergent series?
— Grandi's series and divergent series
practice with this topic
Set the timer (5-30 min), take 20 seconds of prep if you like, start talking. Jot your thoughts onto the sticky-note board.
similar topics
- Roll a circle along a line, and a point on its rim traces not a straight path but a curved arch. The fastest way to slide down between two points is not the straight line but this arch flipped upside down. Why is the quickest path not the straightest one?
- A line contains infinitely many points, and so does a square. But the two contain exactly the same number of points. How can a one-dimensional line be 'equally full' as a two-dimensional square?
- You cannot paint the inside surface of an infinitely long horn, because that would take infinite paint; yet you can fill the same horn completely with liquid paint, because its volume is finite. How can one surface be both unpaintable and fillable?
- Add up the reciprocals of the natural numbers (1 + 1/2 + 1/3 + ...) and the sum grows to infinity; add up only the reciprocals of the primes and it still grows to infinity. Primes keep thinning out, yet they are enough to overflow the sum. How can such 'rare' numbers be this powerful?
- The digits of pi run on forever without repeating; your birth date, your phone number, and quite possibly the name of your unborn grandchild are hiding somewhere in there. How can a pattern that never repeats contain everything?