Author Archives: John

How I got my networked Brother printer to work with i386 drivers on a Debian Jessie x86/amd64 system

Specifically, I’m running Debian 8 (“Jessie”) with kernel 3.16.0-4-amd64 on a desktop computer connected to the router via ethernet cable, and our printer connects wirelessly. The printer is actually a multi-function device, Brother MFC-J835DW. As a prerequisite to printing from … Continue reading

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If a divides b and a divides c, then a divides (b-c)

In reading about Euclid’s proof of the infinitude of prime numbers, the only part that wasn’t completely clear to me was this: If \(p\) divides \(P\) and \(q\), then \(p\) would have to divide the difference of the two numbers, … Continue reading

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Prove that a geometric sequence converges to 0 using Bernoulli’s inequality

Here is a good problem from my first exam in Advanced Calculus (introductory real analysis) taught by Yuri Ledyaev at Western Michigan University. Prove that \(\lim_{n \to \infty} \frac{2^n}{3^n} = 0\). Proof: This proof uses Bernoulli’s inequality, which states that … Continue reading

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Three-bean salad probability density problem

A recipe for three-bean salad includes three different types of beans, \(A\), \(B\), and \(C\). Let the relative weights (masses) of the three bean varieties in a given batch of salad be represented by \(X\), \(Y\), and \(Z\), respectively, such … Continue reading

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A tricky joint probability density problem

Here is problem 7.1.9 from my current probability & statistics textbook, Probability and Statistical Inference by Bartoszynski and Bugaj, which I’m using in the Master’s-level Statistical Theory class taught by Dr. Bugaj herself at Western Michigan University: Variables \(X\) and … Continue reading

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The internet is the best

Specifically, Stack Exchange is the best. I have an mp3 collection of Vivaldi’s complete works, which I acquired by…certain unspecified means, and all the mp3 files were arranged in sub-directories and sub-sub-directories, which is kind of a pain in the … Continue reading

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My recent computer upgrade and troubleshooting

This May, I upgraded the motherboard, CPU, and RAM of my 7-year-old computer and my machine got stuck in a reboot loop. It always POSTed successfully (though sometimes it beeped twice) and loaded the motherboard’s splash screen, but rebooted right … Continue reading

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Proving that a particular sequence is a Cauchy sequence

Here is one of my favorite homework problems from my Advanced Calculus (introductory real analysis) class at Western Michigan University. It is problem 7 from Chapter 1.6 of Advanced Calculus: Theory and Practice by John Petrovic. Let \( 0 < … Continue reading

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My (possibly) favorite episode of The Simpsons: “I Love Lisa”

If I “had” to choose a favorite episode of The Simpsons, I wouldn’t, because about five would be tied as my favorites. Off the top of my head, I’d choose “I Love Lisa”, “Twenty-Two Short Films About Springfield”, “Lemon of … Continue reading

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More Simpsons trivia team names

I occasionally peruse the team names at the Woo Hoo! Classic Simpsons Trivia page because…what better way to spend my free time during my work breaks and such? Here are my recent favorites: Uh, Dan, sir, people are becoming a … Continue reading

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Proof that if f and g are continuous functions, then f/g is also continuous (as long as g(x) ≠ 0)

In almost any calculus or analysis textbook, in the chapter on continuity of functions, you’ll encounter four theorems about the operations on functions that preserve continuity: multiplying a continuous function by a scalar (real number), adding two continuous functions, multiplying … Continue reading

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Convergence of a difficult integral using the limit comparison test

Here’s a great problem from an exam in my second-semester Advanced Calculus (introductory real analysis) course taught by Yuri Ledyaev at Western Michigan University: Find the values of for which the integral converges: $$ \int_{1}^{\infty} \frac{\left(\tan\frac{1}{x}\right)^p}{x+x^2} $$ To determine what … Continue reading

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My favorite Simpsons trivia team names

I’ve wasted several hours this week reading through the names of teams at the classic Simpsons trivia nights that are held in certain restaurants in Chicago, Vancouver, Brooklyn, Toronto, and Hamilton, Ontario. Many of them are hilariously clever. Naturally, I … Continue reading

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Fascinating result of the Intermediate Value Theorem

This is problem #1 from chapter 3.9 in Advanced Calculus: Theory and Practice, my introductory real analysis textbook at Western Michigan University: Suppose that is continuous on and . Prove that there exist , such that and . Informally, this … Continue reading

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Proofs of some trigonometric identities

Remember all those trigonometric identities in the front cover of your calculus book that were too hard to memorize and you didn’t have to anyway? Not the simple ones like \(\sin^2 x + \cos^2 x = 1\) or \(\tan^2 x … Continue reading

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Proof that the limit as n approaches infinity of n^1/n = 1 (\(\lim_{n \to \infty} n^{1/n} = 1\))

Here’s an important limit from real analysis that gives quite a few people, including myself, a lot of trouble: $$ \lim_{n \to \infty}n^{1/n} = 1 $$ Here is the proof that my Advanced Calculus professor at Western Michigan University, Yuri … Continue reading

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Interesting limit from real analysis: lim n!/n^n

In my Advanced Calculus (introductory real analysis) course at Western Michigan University, Dr. Ledyaev gave us this limit as a bonus homework problem to turn in: $$ \begin{align} \lim_{n \to \infty}\frac{n!}{n^n} = ~? \end{align} $$   The answer is that … Continue reading

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Cool theorem about midpoints and parallel vectors from multivariable calculus

This is a cool theorem from multivariable calculus that my professor at Western Michigan University, Steve Mackey, showed us during lecture one day early in the semester. Theorem: Let , , , and be any four points in . Let … Continue reading

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For some reason, I really liked The Host

Last weekend Kathy and I watched the movie The Host on Netflix. It’s based on the novel by Stephenie Meyer, whose name I just found out has no A’s in it. This movie is yet another example of why you … Continue reading

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Probability problem from Star Trek: The Next Generation

In the first episode of season 7 of Star Trek: TNG, “Descent, part II”, a certain character (no spoilers from me!) tells another character that a medical experiment has a 60% chance of failing, meaning it will kill the subject. … Continue reading

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