# 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.

Notes: Sequences and Series is part of One-Variable Calculus. We felt that since this topic had so many capsules associated with it, we wanted to introduce sub-topics. Also, the Number Theory collection of capsules does not correspond to a course in Course Communities, but has topics selected by the Editorial Board for Classroom Capsules and Notes.

## Featured Items

##### Tennis (and Volleyball) without Geometric Series

Solving an expected value problem without using geometric series

##### Quadratic Residues and the Frobenius Coin Problem

An odd prime $$p$$ has $$(p-1)/2$$ quadratic residues mod $$p$$, and for relatively prime $$p$$ and $$q$$ there are $$(p-1)(q-1)/2$$ non-representable Frobenius numbers. The author discusses a relationship between quadratic residues and the Frobenius numbers that accounts for the presence of $$(p-1)/2$$ in both expressions.

##### Proof Without Words: The Area of a Right Triangle

A visual determination of the area of a right triangle is given using an inscribed circle.

##### Inequalities of the Form $$f(g(x)) \geq f(x)$$

The author gives two applications of a method for finding a function $$g$$ such that $$f(g(x)) \geq f(x)$$.

##### Minimum Integral Drawings of the Platonic Graphs

The authors discuss drawing Platonic solids.

##### Another Look at Sylow's Third Theorem

The author uses Moebius inversion to prove Sylow's Third Theorem.