The following series can be shown to converge, but exactly what does it converge to? Euler was supposed to have proven it to sum to pi^2/6, but how? 1 + 1/4 + 1/9 + 1/16 + ... + 1/(r^2) as r -> infinity The following is a small maths puzzle that was asked in another forum, but which I know of no answer to: Assume there exists an omnipotent being who decides to play a game with a lamp. After a minute has passed, he switches it on, 1/2 a min after this, he turns it off, 1/4 of a min later he switches on, 1/8 min later off, 1/16 min later on... After an arbitrarily long time period (1 hour for example, is the lamp switched on or off? What state would it be in?