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Browse Classroom Capsules and Notes

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Displaying 11 - 20 of 67

Another Way to Graph a Sequence

Identifying reasonable sample of points to understand the tail of a sequence

Neither a Worst Convergent Series nor a Best Divergent Series Exists
A short geometric discussion of the convergent and divergent series is given.
Divergence of the Harmonic Series by Rearrangement

A quick proof of the divergence of the harmonic series

Some Sums of Some Significance

The authors discuss certain sums and series that arise in combinatorics and their connections to Sterling numbers.

Natural Logarithms via Long Division

The author finds series expansion for \(\ln 3\) and \(\ln 4\) similar to the alternating harmonic series for \(\ln 2\)..

The Series for \(e\) via Integration
One may derive the series for \(e\) via integration by parts.
Convergence-Divergence of \(p\)-Series
The author shows how to decide which \(p\)-series converge without use of the integral test.
Proofs Without Words Under the Magic Curve

The "magic curve" is \(y=1/x\). Various calculus facts are shown by illustration using Riemann sums for the areas of portions of this curve.

The Alternating Harmonic Series
The author provides an elementary proof that the alternating harmonic series converges to \(\ln 2\).
Divergence of Series by Rearrangement

Assume that a series with non-negative terms converges to \(S\). If from this assumption, a rearrangement of the series can be shown to converge to a different value \(S'\), then the...

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