# 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

##### A Graph Theoretic Summation of the Cubes of the First $n$ Integers

A combinatorial proof of the sum of the cubes of the first $n$ integers is presented, by counting edges in complete bipartite graphs.

##### A Matrix Proof of Newton's Identities

Newton’s identities relate the coefficients of a polynomial to sums of powers of its roots.  The author uses the Cayley-Hamilton theorem and properties of the trace of a matrix to derive Newton’s identities.

##### A Numerical Introduction to Partial Fractions

The author gives a numerical analogue to partial fractions.

##### Image Reconstruction in Linear Algebra

The main object is to solve the inverse problem of recovering the original scene, represented by a vector or a matrix, from its photograph, represented by a product of a matrix and the original vector or matrix. The solution of the resulting matrix equation gives rise to the reconstruction of the original scene.

##### Probabilistic Repeatability Among Some Irrationals

In the decimal expansion of an irrational square root, not all digits are equally likely to appear.

##### Teaching Tip: How $\tan x$ Grows

A mysterious pattern when $\tan x$ tends to infinity is found and explained in terms of L'Hospital's rule.