Browse Classroom Capsules and Notes

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Rearranging the Alternating Harmonic Series

The article gives specific rearrangements of the alternating harmonic series.

The Geometric Series in Calculus

The author argues that many topics in differential and integral calculus could be better approached by an appropriate use of geometric series.

A Calculus Exercise For the Sums of Integer Powers

The author uses differentiation to obtain a standard formula for sums of the form \(1^k + 2^k + \ldots + n^k\), where \(n\) and \(k\) are positive integers.

The Relation Between the Root and Ratio Tests

For series with non-zero terms, the Root Test is interpreted in terms of the geometric mean of the ratios of successive terms (in absolute value). The author also consders the test obtained by...

The Telescoping Series in Perspective

The author describes an application of the telescoping series, \( \sum_{k=1}^{\infty} \frac{1}{k(k+1)}\), to the visual theory of perspective.

Math Bite: Convergence of \(p\) -series

The author gives an elementary proof of the convergence of p-series for p > 1.

The Euler-Maclaurin Formula and Sums of Powers

Bounds are obtained for the sums of powers of the first \(n\) integers.

The Comparison Test — Not Just for Nonnegative Series

It is shown that there are conditions in which comparison tests apply to series with negative terms.

Two Methods for Approximating \(\pi\)

The author summarizes a variety of ways to approximate \(\pi\).

Proof Without Words: \(1/4 + (1/4)^2 + (1/4)^3 + \ldots = 1/3\)

Visual proof of a geometric series identity