# 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

##### Demystifying the Projective Plane

The author describes explicitly the projective plane as a surface formed from a square by joining each pair of opposite edges in a definite way.

##### A Short Proof of Chebychev's Upper Bound

The authors provide a proof, accessible to beginning number theory students, of Chebychevs upper bound on the number of primes no greater than n.

##### Proof without Words: Alternating Sums of Odd Numbers

The author gives a visual proof of the fact that the sum of (2k-1)(-1)n-k from k = 1 to n is n.

##### Dirichlet and Fresnel Integrals via Iterated Integration

A theorem justifying iteration of improper integrals for which Fubinis Theorem does not apply is presented.

##### A Symmetry Criterion for Conjugacy in Finite Groups

If two elements in a group are positioned symmetrically about the main diagonal line in the multiplication table, then the two elements are conjugate.

##### Investigating Possible Boundaries Between Convergence and Divergence

Once students master the Integral Test, it is useful to show that there cannot be a series on the boundary between convergence and divergence.