Discovering Mathematics: A Problem-Solving Approach to by Jiří Gregor

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By Jiří Gregor

Discovering arithmetic: A Problem-Solving method of research with Mathematica and Maple presents a optimistic method of mathematical discovery via cutting edge use of software program know-how. Interactive Mathematica and Maple notebooks are critical to this books’ application as a realistic device for studying. Interrelated suggestions, definitions and theorems are hooked up via links, guiding the reader to numerous dependent difficulties and highlighting a number of avenues of mathematical reasoning.

Interactivity is extra better throughout the supply of on-line content material (available at, demonstrating using software program and in flip expanding the scope of studying for either scholars and academics and contributing to a deeper mathematical realizing.

This publication will attract either ultimate 12 months undergraduate and post-graduate scholars wishing to complement a arithmetic path or module in mathematical problem-solving and research. it is going to even be of use as complementary interpreting for college kids of engineering or technological know-how, and people in self-study.

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Extra info for Discovering Mathematics: A Problem-Solving Approach to Mathematical Analysis with MATHEMATICA® and Maple™

Example text

E. Fn+2 = Fn+1 + Fn . (i) If gcd(Fm+1 , Fm ) = s > 1 then all Fn should be divisible by s, which is evidently wrong. PI 52 1. For any given integer m divide the interval (0, 2π) into m subintervals of equal length 2π/m. 2. Considering numbers k mod 2π , k = 1, 2, . . , m + 1, show that two of them belong to the same subinterval of length 2π/m. 3. Since π is not rational, all integer multiples of the difference of these two numbers (still mod 2π ) form a set, which has the following property: The distance from any point of the interval (0, 2π) to some point of this set is at most 1/m.

Periodicity in the behavior of various dynamical systems is the theme of P 14 – P 24. Compare results obtained for linear differential and difference equations. It is also recommended to formulate and solve actual examples for each of the problems. P 20 gives a simple outlook to the vast area of periodical motions of nonlinear systems. – Some basic examples of Fourier series are given in P 09, P 10, P 11. – More involved cases of the periodicity concept are dealt with in P 08, P 25, P 26, P 27 and periodicity of two variable sequences is a topic in P 28 – P 36.

Ii) What if only one of them is convergent? (iii) Can it happen that none of them is convergent? Concepts 35 Hint: Recall the basic theorems. I 08 •• ↓ PI 08 Consider a sequence {xn } with limn→∞ xn = A and the sequence yn = Show that limn→∞ yn = A. I 09 • ↓ I 09 x1 +x2 +···+xn . n ↑ I 08 Consider a sequence {xn } of positive numbers with limn→∞ xn = A and the sequences zn = √ n x 1 x 2 . . xn , wn = n 1 x1 + 1 x2 + ··· + 1 xn . Find limn→∞ zn and limn→∞ wn , if A > 0. Hint: Consider log zn and I 10 •• 1 wn .

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