Category Archives: calculus

Finding the Area of an Irregular Polygon

Posted on December 18, 2019 by mzspivey

Finding the area of an irregular polygon via geometry can be a bit of a chore, as the process depends heavily on the shape of the polygon.  It turns out, however, that there’s a formula that can give you the … Continue reading →

Posted in analytic geometry, calculus, Green's Theorem | 1 Comment

Six Proofs of a Binomial Identity

Posted on August 22, 2019 by mzspivey

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 →

Posted in binomial coefficients, calculus, combinatorics, generating functions, probability | 2 Comments

Zeno’s Paradoxes

Posted on July 25, 2019 by mzspivey

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 →

Posted in calculus, infinity, logic, paradoxes, real analysis, sequences and series | Leave a comment

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

Posted on November 13, 2017 by mzspivey

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 →

Posted in calculus, real analysis | Leave a comment

A Bonus Question on Convergent Series

Posted on January 6, 2017 by mzspivey

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 →

Posted in calculus, sequences and series | 6 Comments

An arcsecant/arctangent integral

Posted on November 5, 2015 by mzspivey

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

Posted on August 11, 2015 by mzspivey

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 →

Posted in binomial coefficients, calculus, combinatorics | Leave a comment

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

Posted on January 7, 2015 by mzspivey

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 →

Posted in calculus, sequences and series | 2 Comments

Euler Sums, Part I

Posted on December 14, 2013 by mzspivey

Sums of the form , where (i.e., the nth harmonic number of order p) are sometimes called Euler sums.  There are a large number of interesting ways to evaluate these Euler sums, such as converting them to integrals, applying complex analysis, … Continue reading →

Posted in calculus, euler sums, real analysis, sequences and series | 1 Comment

The Product and Quotient Rules for Differential Calculus

Posted on October 25, 2013 by mzspivey

A couple of weeks ago one of my senior colleagues subbed for me on the day we discussed the product and quotient rules for differential calculus.  Afterwards he told me that he had never seen the way that I introduced … Continue reading →

Posted in calculus | Leave a comment