Category Archives: calculus

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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Euler Sums, Part I

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

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The Product and Quotient Rules for Differential Calculus

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

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