Solving an expected value problem without using geometric series

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Keep the December *Monthly* in mind for your holiday reading needs. Does 23/67 equal 33/97? David Pengelley shows us this month that the answer might not be as easy as we think. If you have enjoyed past articles in the *Monthly* about the arbelos, you'll be pleased to learn that Jonathan Sondow introduces us this month to the parbelos, a parabolic analog of the arbelos. James and Michael Henle review *Analysis Through Modern Infinitesimals* by Nader Vakil, and, as always, our Problem Section will keep you thinking.

Stay tuned for January as Mel Nathanson brings in 2014 with a bang with a paper titled "Additive Systems and a Theorem of de Bruijn." Happy New Year from the *American Mathematical Monthly*. —*Scott Chapman*

Vol. 120, No. 10, pp.867-967.

To read the full articles, please log in to the member portal by clicking on 'Login' in the upper right corner. Once logged in, click on 'My Profile' in the upper right corner.

David Pengelley

How might we determine in practice whether 23/67 equals 33/97? Is there a quick alternative to cross-multiplying?

How about reducing? Cross-multiplying checks equality of products, whereas reducing is about the opposite, factoring and cancelling. Do these very different approaches to equality of fractions always reach the same conclusion? In fact, they wouldn’t, but for a critical prime-free property of the natural numbers more basic than, but essentially equivalent to, uniqueness of prime factorization.

This property has ancient, though very recently upturned, origins, and was key to number theory even through Euler’s work. We contrast three prime-free arguments for the property, which remedy a method of Euclid, use similarities of circles, or follow a clever proof in the style of Euclid, as in Barry Mazur’s essay [22].

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.120.10.867

Alexander P. Campbell and Daniel Daners

The resolvent $$(\lambda I-A)^{-1}$$ of a matrix $$A$$* *is naturally an analytic function of $$\lambda\in\mathbb{C}$$, and the eigenvalues are isolated singularities. We compute the Laurent expansion of the resolvent about the eigenvalues of $$A$$. Using the Laurent expansion, we prove the Jordan decomposition theorem, prove the Cayley–Hamilton theorem, and determine the minimal polynomial of $$A$$. The proofs do not make use of determinants, and many results naturally generalize to operators on Banach spaces.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.120.10.877

Olle Häggström and Johan Wästlund

Flip a coin repeatedly, and stop whenever you want. Your payoff is the proportion of heads, and you wish to maximize this payoff in expectation. This so-called Chow–Robbins game is amenable to computer analysis, but while simple-minded number crunching can show that it is best to continue in a given position, establishing rigorously that stopping is optimal seems at first sight to require “backward induction from infinity”.

We establish a simple upper bound on the expected payoff in a given position, allowing efficient and rigorous computer analysis of positions early in the game. In particular, we confirm that with 5 heads and 3 tails, stopping is optimal.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.120.10.893

Will Traves

We prove a generalization of both Pascal’s Theorem and its converse, the Braikenridge–Maclaurin Theorem: If two sets of $$k$$ lines meet in $$k^{2}$$ distinct points, and if $$dk$$ of those points lie on an irreducible curve $$C$$ of degree $$d$$, then the remaining $$k(k-d)$$ points lie on a unique curve $$S$$ of degree $$k-d$$. If $$S$$ is a curve of degree $$k-d$$ produced in this manner using a curve $$C$$ of degree $$d$$, we say that $$S$$ is $$d$$-constructible. For fixed degree $$d$$, we show that almost every curve of high degree is not $$d$$-constructible. In contrast, almost all curves of degree 3 or less are $$d$$-constructible. The proof of this last result uses the group structure on an elliptic curve and is inspired by a construction due to Möbius. The exposition is embellished with several exercises designed to amuse the reader.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.120.10.901

Roger E. Howe

Following A. Cuoco, a proof of the Law of Cosines is given using squares constructed on sides of the triangle. Then the configuration consisting of the original triangle and the triangle defined by the centers of these squares is studied, and several remarkable relationships are found.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.120.10.916

Ulrich Abel

We present a generalization of the Leibniz Rule. One of its consequences is the celebrated Abel identity.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.120.10.924

Jonathan Sondow

The *arbelos* is a classical geometric shape bounded by three mutually tangent semicircles with collinear diameters. We introduce a parabolic analog, the *parbelos*. After a review of the parabola, we use theorems of Archimedes and Lambert to demonstrate seven properties of the parbelos, drawing analogies to similar properties of the arbelos, some of which may be new.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.120.10.929

G. J. O. Jameson

A short proof is given of a comprehensive system of inequalities for $$\Gamma(x+y)/\Gamma(x)$$ including results obtained independently by Artin, Wendel, and Gautschi. Applications include inequalities for binomial coefficients and the Bohr–Mollerup theorem.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.120.10.936

Problems 11740-11746

Solutions 11598, 11610, 11617, 11618, 11619, 11622, 11627

To purchase from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.120.10.941

*Real Analysis Through Modern Infinitesimals*, by Nader Vakil

Reviewed by James M. Henle and Michael G. Henle

JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.120.10.949