# 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

##### An Elegant Mode for Determining the Mode

A technique is introduced for the purpose of finding the relative maximum values of the probability density function. The crux is to find the critical numbers of the exponential of the function instead of the function itself. This results in the same points with much simpler procedures. The same technique can be introduced in lectures of finding critical values in single-variable calculus courses.

##### Fermat's Last Theorem for Gaussian Integer Exponents

Andrew Wiles proves that Fermat's Last Theorem is false for integer exponents larger than $2$.  Using the Gelfond-Schneider Theorem on transcendental numbers, the author generalizes Wiles' result easily by showing that Fermat's Last Theorem is false for  Gaussian integer exponents that are not real.

##### Sums of Integer Powers via the Stolz-Cesàro Theorem

The author uses the Stolz-Cesàro theorem to compute the sums of the integer powers.

##### A New and Improved Method for Finding the Center of Gravity of a Quadrilateral

A new approach to finding the center of gravity for quadrilaterals

##### Proof Without Words: Fibonacci Tiles

The author gives geometrical proofs of a number of identities for the Fibonacci numbers.

##### Divergence of Series by Rearrangement

Assume that a series with non-negative terms converges to $S$. If from this assumption, a rearrangement of the series can be shown to converge to a different value $S'$, then the original series must have diverged. The author uses this result to show that a number of series diverge.