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: Sheng Li (gmachine1729) wrote,
2017-07-04 04:44:00

Sheng Li
gmachine1729
2017-07-04 04:44:00

A cute find closed form of sum problem

Originally published at 狗和留美者不得入内. You can comment here or there.

19692339_10155494787829320_1775528957_n

A friend pinged me this on Facebook. I decided to look at it to exercise my technical chops. Well, the value of the denominator is given by the hint. In the sum of the first $n$ triangular numbers, $k$ is summed $n+1-k$ times, and the number of ways to split $n+1$ items in a line and pick one on each side of the split is the same as the number of ways to select $3$ items from $n+2$ , with the middle one representing the split point. Finally do a partial fractions to telescope. You’ll get $\frac{1/2}{n} - \frac{1}{n+1} + \frac{1/2}{n+2}$ .

Tags: 数学奥赛

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A cute find closed form of sum problem

How weak convergence and weak topology arises naturally from infinite dimensional vector spaces

Using the Minkowski functional to prove separation of sets via a hyperplane

On the Baire category theorem and the open mapping theorem

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