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Author Archives: mzspivey
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
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A Sequence of RightHand 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 righthand Riemann sums formed from equallyspaced partitions to define … Continue reading
Posted in calculus, real analysis
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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 textbased; there are no graphics and no symbolic manipulation (e.g., no solving of … Continue reading
Posted in games, teaching
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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
Posted in number theory, proof techniques
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The Divergence of the Harmonic Series
The fact that the harmonic series, , diverges has been known since the time of Nicole Oresme in the 14th century, but this fact is still somewhat surprising from a numerical standpoint. After all, each successive term only adds a … Continue reading
Posted in harmonic numbers, sequences and series
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The Secretary Problem With the Two Best
The secretary problem is the following: Suppose a manager wants to hire the best person for his secretary out of a group of n candidates. He interviews the candidates one by one. After interviewing a particular candidate he must either (1) … Continue reading
Posted in optimization, probability
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Some Thoughts on Teaching Math for Social Justice
Short version: I’m not in favor of it. For a teacher to use their position of power to push their political and social views on their students is an abuse of that power. Long version: See the rest of this … Continue reading
Posted in campus issues, politics, social justice, teaching
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