fun sums

Some 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.

\lim_{k\to\infty}\prod_{i=2}^k\left(1-\dfrac{1}{i}\right) = \lim_{k\to\infty} \dfrac{1}{k} = 0

For the second method, Friend 1 takes ½² of the cake, Friend 2 takes ⅓² of what remains, and so on. After infinitely many friends take their respective pieces, you get whatever is left.

\lim_{k\to\infty}\prod_{i=2}^k\left(1-\dfrac{1}{i^2}\right) = \lim_{k\to\infty}\dfrac{k+1}{2k} = \dfrac{1}{2}

For the third method, Friend 1 takes ½² 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.

\lim_{k\to\infty}\prod_{i=2}^k\left(1-\dfrac{1}{4i^2}\right) = \lim_{k\to\infty}\dfrac{4(2k+1)}{3\pi k} = \dfrac{2}{\pi}

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