# Classroom Capsules and Notes

Capsules By Courses. We are organizing the capsules into courses, when possible using the same topics as are used in Course Communities. So far we have organized capsules for the following courses:

You may select topics within each course.

## Featured Items

##### The Number-Pad Game

A number game and the discussion of probabilities of winning

##### Pythagorean Triples with Square and Triangular Sides

Describing Pythagorean Triples with one square side and a triangular side. Their number is infinite.

##### On Antiderivatives of the Zero Function

The author discusses an intuitive direct proof of the fact that functions with zero derivative must be constant, which turns into a rigorous proof by simply invoking the completeness of the real numbers.

##### Coefficients of the Characteristic Polynomial

The author derives formulas for the coefficients of the characteristic polynomial of a square matrix in terms of the partial derivatives of its determinant, which is viewed as a function of the entries of the matrix.

##### Euler Convergence: Probabilistic Considerations

The authors show, using probabilistic methods, that convergence implies Euler convergence, but not conversely.

##### Approaching $\ln x$

The integral $\int_{1}^{x} t^{-1+1/n} dt$ is shown to converge to $\ln x$.