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the 3 X loop

Posted in Mountains, pictures, Running, Travel with tags Annecy, cross, French Alps, Hautes Alpes, jatp, Lac d'Annecy, lake, outdoor swimming, rain, sunrise, vacations on June 6, 2021 by xi'an

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Lac d’Annecy [jatp]

Posted in Statistics with tags Annecy, French Alps, Hautes Alpes, hiking, jatp, Lac d'Annecy, lake, Mont Veyrier, vacations on May 31, 2021 by xi'an

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fun sums

Posted in Books, Kids, Statistics with tags limits, mathematical puzzle, riddle, Stirling approximation, sums, The Riddler, vacations on May 26, 2021 by xi'an

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 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.

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

For the third method, Friend 1 takes 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.

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