Category Archives: Math

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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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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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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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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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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The infinitude of prime numbers—Euclid’s proof in my own words

Euclid is believed to be the first mathematician to prove that there are infinitely many prime numbers. Most of us learn only that Euclid established and codified the framework of two- and three-dimensional geometry, but he accomplished far more than … Continue reading

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The coins in the dark puzzle

In the New York Times’ Numberplay column yesterday, Gary Antonick presents an old but good logic puzzle: There are twenty-six coins lying on a table in a totally dark room. Ten are heads and sixteen are tails. In the dark … Continue reading

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