An Exceptional Exponential Function
We show that there is a link between a standard calculus problem of finding the best view of a painting and special tangent lines to the graphs of exponential functions. Surprisingly, the exponential function with the “best view” is not the one with the base e. A similar link is established for families of functions obtained by composing exponential functions with a fixed linear function. The key tool in the proof is the Lambert W function.
More Designer Decimals: The Integers and their Geometric Extensions
O-Yeat Chan and James Smoak
The fraction 10000/9801 has an intriguing decimal expansion, namely 1.02030405... In this paper, we investigate the properties of this fraction via an arithmetical approach. The approach also yields a class of fractions whose decimal expansions involve higher-dimensional analogues of the integers, the n-dimensional pyramidal numbers, thereby showing both arithmetic and geometric connections.
The Divergence of Balanced Harmonic-like Series
Carl V. Lutzer and James E. Marengo
Consider the series where the value of each an is determined by the flip of a coin: heads on the nth toss will mean that an =1 and tails that an = -1 . Assuming that the coin is “fair,” what is the probability that this harmonic-like series converges? After a moment’s thought, many people answer that the probability of convergence is 1. This is correct (though the proof is nontrivial), but it doesn’t preclude the existence of a divergent example. Indeed, Feist and Naimi provided just such an example in 2004. In this paper, we construct an uncountably infinite family of examples as a companion result.
An Interview with H. W. Gould
Scott H. Brown
In this interview, one of the world’s foremost combinatorialists, H.W. Gould, shares some of the experiences of his early life as well as his lengthy career of more than forty-seven years at the University of West Virginia.
Fallacies, Flaws, and Flimflam
Ed Barbeau, editor
Ricardo Alfaro and Steven Althoen, editors
Another Look at Some p-Series
A geometric construction is provided that simultaneously shows why the p-series diverges for p = 1, yet converges for p = 2 and 3.
Summing Cubes by Counting Rectangles
Arthur T. Benjamin, Jennifer L. Quinn, and Calyssa Wurtz
The starting point of this capsule is counting the number of rectangles in an m × n checkerboard (with a neat four-line derivation). From this, the formula for the sum of the first n cubes of positive integers and other results are derived..
The Converse of Viviani’s Theorem
Zhibo Chen and Tian Liang
Viviani’s Theorem, discovered over 300 years ago, states that inside an equilateral triangle, the sum of the perpendicular distances from a point P to the three sides is independent of P. In this capsule Chen and Liang establish its converse in a stronger version.