This paper provides a detailed outline of a mathematical research exploration for use in an introductory high school or college Calculus class and is directed toward teachers of such courses. The discovery is accompli...This paper provides a detailed outline of a mathematical research exploration for use in an introductory high school or college Calculus class and is directed toward teachers of such courses. The discovery is accomplished by introducing a novel method to generate a polynomial expression for each of the Euler sums, ΣNk=0kn,n∈Z+ . The described method flows simply from initial discussions of the Riemann sum definition of a definite integral and is readily accessible to all new calculus students. Students investigate the Bernoulli numbers and the interesting connections with Pascal's Triangle. Advice is offered throughout as to how the project can be assigned to students and offers multiple suggestions for additional exploration for any motivated student.展开更多
Let △（x）be the error term in the asymptotic formula for the counting function of square-full integers.In the present paper it is proved that △（x）=O(x<sup>27/4+ε</sup>),which improves on the exponent...Let △（x）be the error term in the asymptotic formula for the counting function of square-full integers.In the present paper it is proved that △（x）=O(x<sup>27/4+ε</sup>),which improves on the exponent 33/5 obtained by X.D.CAO.展开更多
Can a person chase up a turtle? This might sound like a very stupid question. However, this question had been disturbing mathematicians for many centuries. It is possible to solve the paradox of Achilles and the turtl...Can a person chase up a turtle? This might sound like a very stupid question. However, this question had been disturbing mathematicians for many centuries. It is possible to solve the paradox of Achilles and the turtle with only high school knowledge. This is the beauty of calculus, the concept that mathematicians struggled to invent and improve. In more than 2000 years, mathematicians tried different ways to study and invent calculus, and the invention finally led to a revolution in the world of math.展开更多
文摘This paper provides a detailed outline of a mathematical research exploration for use in an introductory high school or college Calculus class and is directed toward teachers of such courses. The discovery is accomplished by introducing a novel method to generate a polynomial expression for each of the Euler sums, ΣNk=0kn,n∈Z+ . The described method flows simply from initial discussions of the Riemann sum definition of a definite integral and is readily accessible to all new calculus students. Students investigate the Bernoulli numbers and the interesting connections with Pascal's Triangle. Advice is offered throughout as to how the project can be assigned to students and offers multiple suggestions for additional exploration for any motivated student.
基金Project supported by the National Natural Science Foundation of China
文摘Let △（x）be the error term in the asymptotic formula for the counting function of square-full integers.In the present paper it is proved that △（x）=O(x<sup>27/4+ε</sup>),which improves on the exponent 33/5 obtained by X.D.CAO.
文摘Can a person chase up a turtle? This might sound like a very stupid question. However, this question had been disturbing mathematicians for many centuries. It is possible to solve the paradox of Achilles and the turtle with only high school knowledge. This is the beauty of calculus, the concept that mathematicians struggled to invent and improve. In more than 2000 years, mathematicians tried different ways to study and invent calculus, and the invention finally led to a revolution in the world of math.
