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Mathematics Magazine - December 2016

This issue has holiday gifts for everyone. Jenny’s number (867-5309) and pi make appearances in articles on nested radicals by Devyn Lesher and Chris Lind and by Mu-Ling Chang and Chia-Chin Chang, respectively. Bruce and Robert Torrence examine the dice game Left, Center, Right or LCR, connecting the endgame to Fibonacci and Lucas numbers. Finbarr Holland offers a characterization of quadratic polynomials. Raymond Beauregard and Vladimir Dobrushkin provide a recurrence for the general rational function for well-known polynomial sequences (e.g., Chebyshev, Fibonacci, Lucas, Fermat, et al.) And, Russell Gordon, George Stoica, and Mark Lynch each provide one of a set of analysis articles. There is also an interview with mathematician and digital artist Anne Burns, problems and solutions from the 57th International Mathematical Olympiad, and a crossword in advance of the 2017 Joint Math Meetings.

Michael A. Jones, Editor


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Vol. 89, No. 5, pp. 317 – 400


Convergence Results for the Class of Periodic Left Nested Radicals

p. 319.

Devyn A. Lesher and Chris D. Lynd

We investigate several examples of left nested radicals and prove four convergence theorems. In the proofs of the theorems we provide a recipe for constructing nested radicals with a predetermined end-behavior. We conclude the paper by constructing a nested radical whose computed sequence becomes an endless repetition of the digits in the phone number 867-5309, just like in the song by Tommy Tutone.

To purchase from JSTOR: 10.4169/math.mag.89.5.319

Evaluation of Pi by Nested Radicals

p. 336.

Mu-Ling Chang and Chia-Chin (Cristi) Chang

We use nested radicals to represent pi as a limit.

To purchase from JSTOR: 10.4169/math.mag.89.5.336

Proof Without Words: Factorial Sums

p. 338.

Tom Edgar

We provide a visual proof of a factorial sum identity implying the existence of the factorial base number system.

To purchase from JSTOR: 10.4169/math.mag.89.5.338

Crossword Puzzle

p. 341.

Brendan Sullivan

Fibonacci, Lucas, and a Game of Chance

p. 342.

Bruce Torrence and Robert Torrence

A simple game of chance is introduced, where it is shown that each player’s winning probability is a ratio of Fibonacci and Lucas numbers.

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Characterizations of Quadratic Polynomials

p. 352.

Finbarr Holland

This note was motivated by a student’s misconception that the average velocity over a time interval of a particle in rectilinear motion is the arithmetic mean of its velocities at the ends of the interval. We present an algebraic proof that this property holds only if the particle has constant acceleration, thereby recovering a well-known result. A characterization is also provided of differentiable functions on (−∞,∞) whose restrictions to the intervals (−∞, 0] and [0,∞) are quadratic polynomials. In addition, a calculus-free proof is presented to show that a certain feature of the mean value theorem holds only for quadratic polynomials.

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Proof Without Words: Alternating Row Sums in Pascal’s Triangle

p. 358.

Ángel Plaza

Based on the Pascal’s identity, we visually demonstrate that the alternating sum of consecutive binomial coefficients in a row of Pascal’s triangle is determined by two binomial coefficients from the previous row.

To purchase from JSTOR: 10.4169/math.mag.89.5.358

Powers of a Class of Generating Functions

p. 359.

Raymond A. Beauregard and Vladimir A. Dobrushkin

We consider positive powers of a rational generating function F(x,z) of the variable z whose numerator has degree at most 1, denominator has degree 2, and where coefficients are functions of x. Many well-known polynomial sequences are generated by functions of this form, including the generating function U(x,z) for Chebyshev polynomials Un (x) of the second kind; the square root U(x,z)1/2 generates the sequence of Legendre polynomials. The numerical sequence Un (3) is the sequence of numbers whose squares are triangular numbers, and the square root of its generating function breeds the sequence of central Delannoy numbers. Our goal is to provide a recurrence for the sequence generated by F(x,z)α, where α is a positive real number, thus providing a reasonable way for computing sequential values.

To purchase from JSTOR: 10.4169/math.mag.89.5.359

A Bounded Derivative That Is Not Riemann Integrable

p. 364.

Russell A. Gordon

We present an example, different from Volterra’s, of a bounded derivative that is not Riemann integrable. The existence of such functions was one of the motivations for Lebesgue to devise a stronger integration process. The goal of the presentation is to keep the ideas at a level appropriate for an undergraduate real analysis student.

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Generating Continuous Nowhere Differentiable Functions

p. 371.

George Stoica

In this note we show how can one associate a continuous and highly nondifferentiable function g to any bounded function f on a compact interval.

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A Function With Continuous Nonzero Derivative Whose Inverse Is Nowhere Continuous

p. 373.

Mark Lynch

A function with continuous nonzero derivative whose inverse is nowhere continuous is given. The function satisfies all the premises of the inverse function theorem except one: it is not defined on an open set.

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Anne Burns: Mathematical Botanist

p. 375.

Amy L. Reimann and David A. Reimann

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Problems and Solutions

p. 378.

Proposals, 2006-2010

Quickies, 1065-1066

Solutions, 1976-1980

Answers, 1065-1066

To purchase from JSTOR: 10.4169/math.mag.89.5.378


p. 386.

Topology Nobel; 4-peg Hanoi; unusual statistics texts

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News and Letters

57th International Mathematical Olympiad

p. 388.

Po-Shen Loh

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p. 398.

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