Fibonacci Series from Power Series

We show how every power series gives rise to a Fibonacci series and a companion series involving Lucas numbers. For illustrative purposes, Fibonacci series arising from trigonometric functions, inverse trigonometric functions, the gamma function and the digamma function are derived. Infinite series involving Fibonacci and Bernoulli numbers and Fibonacci and Euler numbers are also obtained.

The chord-length distribution of a polyhedron

The chord-length distribution function [γ′′(r)] of any bounded polyhedron has a closed analytic expression which changes in the different subdomains of the r range. In each of these, the γ′′(r) expression only involves, as transcendental contributions, inverse trigonometric functions of argument equal to R[r, Δ1], Δ1 being the square root of a second-degree r polynomial and R[x, y] a rational function. As r approaches δ, one of the two end points of an r subdomain, the derivative of γ′′(r) can only show singularities of the forms |r − δ|−n and |r − δ|−m+1/2, with n and m appropriate positive integers. Finally, the explicit analytic expressions of the primitives are also reported.

New upper bounds for noncentral chi-square cdf

Some new upper bounds for noncentral chi-square cumulative density function are derived from the basic symmetries of the multidimensional standard Gaussian distribution: unitary invariance, components independence in both polar and Cartesian coordinate systems. The proposed new bounds have analytically simple form compared to analogues available in the literature: they are based on combination of exponents, direct and inverse trigonometric functions, including hyperbolic ones, and the cdf of the one dimensional standard Gaussian law. These new bounds may be useful both in theory and in applications: for proving inequalities related to noncentral chi-square cumulative density function, and for bounding powers of Pearson’s chi-squared tests.

2. Sines, cosines, and their relatives

‘Sines, cosines, and their relatives’ begins by defining the basic trigonometric functions—sine, cosine, and tangent—and explaining their use. These functions are geometric quantities defined using the ratios of the Opposite, Adjacent, and Hypotenuse sides of the right triangle. Less common functions are the cosecant, secant, and cotangent functions. The history of the naming of the trigonometric functions is discussed along with an explanation of even more obscure functions: the versed sine, versed cosine, exsecant, and excosecant. The versed sine was used frequently in practical applications like astronomy, navigation, and surveying. Finally, inverse trigonometric functions and graphs of trigonometric functions are considered.

Robotics Simulators in STEM education

This article discusses STEM (Science, Technology, Engineering and Mathematics) education as an initiative from various countries around the world to address young people's lack of interest in careers in Science, Mathematics, Technology and Engineering. Understanding that STEM education must explore two or more of STEM themes, using transdisciplinarity, engaging the student in activities with this approach, we present the studies of an activity proposal integrating Engineering, Technology and Mathematics with the objective of learning Mathematics. In this activity students work with situations involving robotics and, for solution, use robotic arm simulators, developed in GeoGebra software, that simplify the real environment in which the robotic arm manipulates an object positioned in the plane, taking to organize strategies by identifying and applying mathematics, such as trigonometry with right triangle, trigonometric identities, inverse trigonometric functions, to solve the problem. An experiment was conducted to validate the simulators with undergraduate mathematics students from Universidade Luterana do Brasil (ULBRA) in the city of Canoas in Rio Grande do Sul. The results indicate that it is possible to integrate the STEM areas with the developed simulators, being indicated for activities with high school students (10th or 11th grade).