**S**ome sums and limits found from a [vacation] riddle by The Riddler:

For the first method, Friend 1 takes half of the cake, Friend 2 takes a third of what remains, and so on. After infinitely many friends take their respective pieces, you get whatever is left.

For the second method, Friend 1 takes ½^{²}of the cake, Friend 2 takes ⅓^{²}of what remains, and so on. After infinitely many friends take their respective pieces, you get whatever is left.

For the third method, Friend 1 takes ½^{²}of the cake, Friend 2 takes ¼² of what remains, Friend 3 takes ⅙² of what remains after Friend 2, and so on. After your infinitely many friends take their respective pieces, you get whatever is left.