By Martin Bohner, Allan C. Peterson

First-class introductory fabric at the calculus of time scales and dynamic equations.; a variety of examples and workouts illustrate the various program of dynamic equations on time scales.; Unified and systematic exposition of the subjects permits strong transitions from bankruptcy to chapter.; individuals comprise Anderson, M. Bohner, Davis, Dosly, Eloe, Erbe, Guseinov, Henderson, Hilger, Hilscher, Kaymakcalan, Lakshmikantham, Mathsen, and A. Peterson, founders and leaders of this box of study.; helpful as a accomplished source of time scales and dynamic equations for natural and utilized mathematicians.; accomplished bibliography and index whole this article.

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To do this we have Mathematica make a decision using an If function. Now, we need Mathematica to recognize which of the calculations are fractions, and which are whole numbers, but unfortunately we currently have no idea how we might do this. One possible answer is to use the modular arithmetic calculations from above, remembering that if n/a ∈ N then n ≡ 0 mod a. This is something we already know how to express in Mathematica. In order to see if, say, 3 was a divisor of our 6 then we could issue the command In[169]:= If[6 ~Mod~ 3 == 0, 6 / 3] Out[169]= 2 The above code should be read as “If 6 is equal to 0 modulo 3 then calculate 6/3,” and because 6 is most certainly equivalent to 0 modulo 3, Mathematica has correctly gone on to calculate 6/3 = 2.

In[112]:= N[Infinity] Out[112]= π2 6 Ih either case, there we have it. It looks very much as if the series converges to 1/6 π 2 , provided we trust Mathematica’s limit computation. To verify this analytically we would need to use the integral test, but we shall not do so here. We will see how to perform calculus with Mathematica in Chapter 2, but performing the integral test for this series is left as an exercise to the reader. 2 Loops Until now if we wanted to perform something several times, we either typed it in multiple times at the command prompt, or we constructed a list.

Recall that if n can be divided by another natural number a evenly—that is, n/a is a natural number—we use the notation a|n and say that a divides n or that a is a divisor of n. Furthermore, if a|n then n = ka for some k ∈ N and so, recalling modular arithmetic, n ≡ 0 mod a. The problem we now try to solve now with Mathematica is to ﬁnd all the divisors of a number. To begin with, it is helpful to know that Mathematica can perform modular 24 1 Number Theory arithmetic using the Mod function. Simply put, entering Mod[a, b] will calculate the modulus of a (modulo b).