By Patrick M. Fitzpatrick

ISBN-10: 0534376037

ISBN-13: 9780534376031

Simply grasp the basic suggestions of mathematical research with complicated CALCULUS. offered in a transparent and easy approach, this complicated caluclus textual content leads you to an exact knowing of the topic via supplying you with the instruments you want to be successful. a large choice of workouts is helping you achieve a real knowing of the cloth and examples show the importance of what you examine. Emphasizing the team spirit of the topic, the textual content indicates that mathematical research isn't really a suite of remoted evidence and methods, yet quite a coherent physique of information.

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**Additional resources for Advanced Calculus**

**Example text**

E. The limit of a convergent sequence in the interval (a, b) also belongs to (a, b). 2. Show that the set ( -oo, 0] is closed. 3. Show that every number is the limit of a sequence of irrational numbers. 4. Show that the set of irrational numbers fails to be closed. S. Show that a sequence {an} is bounded if and only if there is an interval [c, d] such that {an} is a sequence in [c, d]. 1 we showed that limn---+oo 1In = 0. It is clear that constant sequences converge to their constant values. Thus, using the sum, product, and quotient properties of convergent sequences, we can combine these two examples to obtain further examples of convergent sequences.

Use Cauchy's Inequality to show that for any numbers a and b and a natural number n, ab ~ ~ (na 2 + ~b2 ). ) 17. Let a, b, and c be nonnegative numbers. Prove the following inequalities: a. ab + be + ca ::: a 2 + b 2 + c2 • b. Babe :=: (a+ b)(b + c)(c +a). c. abc(a + b +c):=: a 2 b 2 + b 2 c 2 + c2 a 2 • TOOLS FOR ANALYSIS 21 18. A function f : IR ---+ IR is called strictly increasing provided that f (u) > f (v) for all numbers u and v such that u > v. a. Define p(x) = x 3 for all x. Prove that the polynomial p: IR ---+ IR is strictly increasing.

Also, show that if a =/: 0, then 1 (1 -a) - = 1 + (1- a)+ (1- a) 2 + - a 3 a 21. Prove that if nand k are natural numbers such that k ::::: n, then 22. Use the formula in Exercise 21 to provide an inductive proof of the Binomial Formula. 23. Let a be a nonzero number and m and n be integers. Prove the following equalities: a. am+n =am an. b. (ab)n = anbn. 24. A natural number n is called even if it can be written as n = 2k for some other natural number k, and is called odd if either n = 1 or n = 2k + 1 for some other natural number k.

### Advanced Calculus by Patrick M. Fitzpatrick

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