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Browse Classroom Capsules and Notes

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Displaying 1071 - 1080 of 1213

A series of rational numbers whose absolute values converge to a rational number while the series itself converges to an irrational number is presented.

The author shows that a square inscribed in a semicircle has \(2/5\) the area of a square inscribed in a circle of the same radius.

The author proposes two extensions of the Monte Hall problem, with solutions involving the numbers \(\pi\) and \(e\), respectively.
A combinatorial proof of the sum of the cubes of the first \(n\) integers is presented, by counting edges in complete bipartite graphs.
The author proves visually that if \(a\) and \(b\) are the legs and h the altitude to the hypotenuse \(c\) of a right triangle, then \((1/a)^2 + (1/b)^2 = (1/h)^2\).
Arguments for certain sums of sines and cosines are given.
This article gives an explanation why the solution to a number partitioning problem would be \(3\). The answer is that \(3\) is close to \(e\).
The article attempts to explain why exponential functions were chosen to define hyperbolic functions.
An elementary proof for the generalized Schwarz inequality.

Cramer's Rule gives an explicit formulation for the unique solution to a system of \(n\) equations in \(n\) unknowns when the coefficient matrix of the system is invertible.  The...