Category Archives: calculus

Six Proofs of a Binomial Identity

I’m a big fan of proving an identity in multiple ways, as I think each perspective gives additional insight into why the identity is true.  In this post we’ll work through six different proofs of the binomial identity . 1. … Continue reading

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Zeno’s Paradoxes

The ancient Greek philosopher Zeno of Elea is known for proposing several paradoxes related to time, space, motion, and infinity.  In this post we’ll focus on one of Zeno’s paradoxes and discuss how some ideas associated with calculus might or … Continue reading

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A Sequence of Right-Hand Riemann Sums Whose Limit Does Not Exist

Recently in my advanced calculus class one of my students asked a (perhaps the) key question when we hit the integration material: Why can’t you just use the limit of the right-hand Riemann sums formed from equally-spaced partitions to define … Continue reading

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A Bonus Question on Convergent Series

Occasionally when teaching the sequences and series material in second-semester calculus I’ve included the following question as a bonus: Question: Suppose  is absolutely convergent.  Does that imply anything about the convergence of ? The answer is that converges.  I’m going … Continue reading

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An arcsecant/arctangent integral

Recently in my integral calculus class I assigned the problem of evaluating . I intended for the students to recognize the integrand as similar to the derivative of arcsecant: This leads to the substitution , with .  The original integral then … Continue reading

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Vandermonde’s Identity from the Generalized Product Rule

Vandermonde’s Identity, is one of the more famous identities involving the binomial coefficients.  A standard way to prove it is with the following combinatorial argument. How many ways are there to choose a committee of size r from a group of n men … Continue reading

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A Simple Proof That the p-Series Converges for p > 1

Proving that the p-series converges for is a standard exercise in second-semester calculus.  It’s also an important property to know when establishing the convergence of other series via a comparison test.  The usual way I do this in class is with the … Continue reading

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