Browse Classroom Capsules and Notes

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A Geometric Look at Sequences That Converge to Euler's Constant

This capsule investigates the sequences that converge to Euler's constant. By utilizing the geometric description of the terms, the author can obtain a rate of convergence comparable to...

Another Look at Some p-series

Certain p-series are the focus of this capsule. This project comes with scenarios to help students "visualize" the convergence or divergence of the p-series.

Proof Without Words: Geometric Series Formula

The result is illustrated for ratio 1/9, via an approach that is readily generalized. In particular, the result \( \frac{1}{n} + \frac{1}{n^2} + \frac{1}{n^3} + \cdots = \frac{1}{n...

Stirred, Not Shaken, by Stirling's Formula

In this note an elementary proof of Stirling's asymptotic formula for \(n!\) is given.

Proof Without Words: The Alternating Harmonic Series Sums to \(\ln 2\)

The result in the title is demonstrated graphically.

Proof Without Words: Mengoli's Series

Mengoli`s Series is presented visually .

\( \lim_{m \rightarrow \infty} \sum_{k=0}^{m} (k/m)^m\) \(= e/(e-1) \)

Two proofs, one elementary, of the limit in the title are presented.

A GM-AM Ratio

The limit of the ratio of the geometric mean to the arithmetic mean of certain sequences is studied, using Riemann sums.

Math Bite: A Magic Eight

The authors show that a certain sequence unexpectedly converges to 8.