Author Archives: mzspivey

Solving Second-Order Difference Equations with Constant Coefficients

Suppose you have a linear, homogeneous second-order difference equation with constant coefficients of the form .  The solution procedure is as follows: “Guess” a solution of the form .  Then substitute this guess into the difference equation to obtain .  … Continue reading

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Adventures in Fine Hall

The Princeton Alumni Weekly has a great article up about the mathematicians at Princeton and the Institute for Advanced Study in the 1930s and 1940s.  I’ve heard a lot of anecdotes about mathematicians over the years, but I had not … Continue reading

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A Quasiperfect Number Must Be an Odd Perfect Square

Let be the sum of divisors function.  For example, , as the divisors of 5 are 1 and 5.  Similarly, , as the divisors of 6 are 1, 2, 3, and 6. A perfect number is one that is equal … Continue reading

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Determinant of a Symmetric Pascal Matrix

The infinite symmetric Pascal matrix Q is given by where entry in Q is .  (Note that we begin indexing the matrix with 0, not 1, in keeping with the way Pascal’s triangle is usually indexed.) The purpose of this post is to … Continue reading

Posted in binomial coefficients, matrices | 2 Comments

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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Mathematics through Narrative: A Beauty Cold and Austere

Earlier this year I spent some time (well, a lot of time) writing a mathematical computer game.  It’s called A Beauty Cold and Austere (ABCA), and it’s text-based; there are no graphics and no symbolic manipulation (e.g., no solving of … Continue reading

Posted in games, teaching | 2 Comments

The Validity of Mathematical Induction

Suppose you have some statement .  Mathematical induction says that the following is sufficient to prove that is true for all natural numbers k. is true. For any natural number k, if is true, then is true. The idea is that the first … Continue reading

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