# 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

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

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

##### Shanille Practices More

A solution to a probabilistic Putnam Exam Problem is presented.

##### On Candido's Identity

The authors discuss the existence of functions from the nonnegative reals to the nonnegative reals that satisfy the functional equation underlying Candido`s identity.

##### Logarithmic Differentiation: Two Wrongs Make a Right

Differentiate $f(x)^{g(x)}$ first as if $g$ was a constant, then as if $f$ was a constant. Presto!

##### The Arithmetic of Algebraic Numbers: An Elementary Approach

If $r$ and $s$ are algebraic numbers, then $r + s$, $rs$, and $r/s$ are also algebraic. The proof provided in this capsule uses the ideas of characteristic polynomials, eigenvalues, and eigenvectors.

##### Proofs Without Words Under the Magic Curve

The "magic curve" is $y=1/x$. Various calculus facts are shown by illustration using Riemann sums for the areas of portions of this curve.