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Browse Classroom Capsules and Notes

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Displaying 61 - 67 of 67

Sums of Integer Powers via the Stolz-Cesàro Theorem

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

Math Bite: A Magic Eight
The authors show that a certain sequence unexpectedly converges to 8.
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.

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

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

Proof Without Words: Mengoli's Series

Mengoli`s Series is presented visually .

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

The result in the title is demonstrated graphically.

Stirred, Not Shaken, by Stirling's Formula

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

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