ARTICLESKirkman’s Schoolgirls Wearing Hats and Walking through Fields of Numbers
When Euler Met l’Hôpital
by William Dunham
In the fifteenth chapter of his 1755 text on differential calculus, Leonhard Euler considered indeterminate expressions of the form 0/0 and provided a string of dazzling examples that make modern treatments of the subject look quite pedestrian. Some indeterminate forms he evaluated with basic algebra, some with trigonometric identities, and some with infinite series. And, as the title suggests, he rolled out the heavy artillery of l’Hôpital’s rule. In Euler’s hands, this served not just to solve specially-concocted problems but to evaluate the sum of the first n whole numbers. More improbably, he applied l’Hôpital’s rule (three times!) to resolve the Basel problem, i.e., to sum the infinite series of the reciprocals of the squares. By examining these results, we hope to reveal a master at work.
The Magic EIGHT
by Paul and Vincent Steinfeld
The sequence 9/1, 98/12, 987/123, 9876/1234, ... with a suitable interpretation, is seen to converge to 8.
Matroids You Have Known
by David L. Neel and Nancy Ann Neudauer
The notion of independence surfaces in several branches of mathematics, even in the undergraduate curriculum. Is there a mathematical theory of independence? We introduce and discuss the theory of matroids, mathematical objects created to characterize independence. We show how matroids can arise in linear algebra and graph theory, as well as in certain applications involving optimization. We discuss their intimate connection to the greedy algorithm, and finish with an application to a scheduling problem.
LETTER TO THE EDITOR
What If Archimedes Had Met Taylor?
by Richard D. Neidinger
The enjoyable article “What if Archimedes Had Met Taylor?” (this MAGAZINE, October 2008, pp. 285-290) can be understood in terms of eliminating error terms. This leads to a different concluding approximation that is more in the spirit of the note by combining previous estimates for improvement.
Series that Probably Converge to One
by Thomas J. Pfaff and Max M. Tran
Infinite series that converge to one are shown to arise from various probability scenarios. For a variety of more and more complicated experiments, we define a collection of disjoint events that exhaust the sample space. The probabilities of these events are terms in an infinite series that sums to one.
Flip the Script: From Probability to Integration
by David A. Rolls
After a first course in probability, what might be said about calculus that couldn't be said before? Probability density functions must integrate to one. Can we use this to advantage? Under independence, the expected value of a product of random variables is the product of the expected values? How might we use this? Reinterpreting certain integrands as recognizable probability density functions, we show how calculating certain integrals can become quite easy. Building on this idea, we then introduce Monte Carlo integration and give some history of the method.
by Inta Bertuccioni
We give a simple proof, based on an idea of Hillel Gauchman, of the existence of infinitely many primes congruent to 1 modulo any fixed positive integer.
PROOF WITHOUT WORDS
An Arctangent Identity
by Hasan Unal
If (x,y) is a point on the unit circle in the first quadrant, then the sum of the arctangent of (1-x)/y and the arctangent of (1-y)/x is π/4
Evil Twins Alternate with Odious Twins
by Chris Bernhardt
An integer is called evil if the number of ones in its binary expansion is even and odious if the number of ones in the binary expansion is odd. If two consecutive integers are evil then this is called a pair of evil twins and if two consecutive integers are odious then this is called a pair of odious twins.
by Ángel Plaza
By comparing slopes of graphs, we prove that ax - 1 > x (a - 1), which is known as Bernoulli's inequality.
Letter to the Editor: Isosceles Dissections
by Roger B. Nelsen
I enjoyed Des MacHale’s “Proof Without Words: Isosceles Dissections” in the December 2008 issue of the Magazine. However, his third claim (a triangle can be dissected into two isosceles triangles if and only if one of its angles is three times another or if the triangle is right angled) is incorrect.
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