Browse Classroom Capsules and Notes

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A Note on Taylor's Series for \(\sin(ax+b)\) and \(\cos(ax+b)\)

Taylor series for sine and cosine functions and their derivatives are discussed.

Neither a Worst Convergent Series nor a Best Divergent Series Exists

A short geometric discussion of the convergent and divergent series is given.

Bounding the Roots of Polynomials

Some bounds on the roots of the polynomial functions are discussed.

Order Relations and a Proof of l'Hôpital's Rule

Leonard Gillman uses a definition of limit based on order rather than metric properties of the real line to prove l'Hôpital's Rule.

Divergence of the Harmonic Series by Rearrangement

A quick proof of the divergence of the harmonic series

Exploiting a Factorization of \(x^n - y^n\)

A new proof of the binomial identity is given and the method is applied to prove other identities.

The Pen and the Barn

In the classical problem of fixed perimeter and maximum area of the calculus, it is shown that the length of the fence affects the shape of the optimal pen.

On Dividing Coconuts: A Linear Diophantine Problem

Two solutions for the classical coconut dividing problem of the five sailors on an island are given.

Halley's Gunnery Rule

The author solves the classical problem of firing a projectile through a target situated on an inclined plane.

Introduction to Limits, or Why Can't We Just Trust the Table?

The author shows with an example that you cannot predict the correct limit by selecting a few values nearby.