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Browse Classroom Capsules and Notes

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Displaying 21 - 30 of 67

The author presents an easy absolute convergence test for series based solely on differentiation, with examples.

A variation of a well-known approximation of \(e\) is analyzed.

This is a short proof of a famous result of Euler about summation of the following series: \(\sum 1/{n^2} = {\pi^2}/6\).

The authors provide several examples of evaluating difficult limits by using Riemann sums. Note that many of these limits are usually solved by Stirling's formula or series methods....

The author provides a concise proof that \(\sum_{n=0}^{\infty}1/(2n+1)^2 ={\pi}^2/8\).

The sum of reciprocals of triangular numbers is computed visually.

Using integration, the author finds the sums of a class of series whose denominators involve bionomial coefficients.

Clever partitioning of a square into three equal regions yields an alternating series summing to 1/3.

The subseries of the harmonic series, \(\sum_{n=1}^{\infty} \frac{1}{n}\), which consists of all terms with one or more nines in the digits of \(n\), is a divergent series.

The author gives a proof of Stirling's formula accessible to the first year calculus students.