Download Advanced Calculus: Theory and Practice by John Srdjan Petrovic PDF

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By John Srdjan Petrovic

Suitable for a one- or two-semester direction, Advanced Calculus: thought and Practice expands at the fabric lined in user-friendly calculus and provides this fabric in a rigorous demeanour. The textual content improves scholars’ problem-solving and proof-writing talents, familiarizes them with the old improvement of calculus recommendations, and is helping them comprehend the connections between diverse topics.

The booklet takes a motivating procedure that makes rules much less summary to scholars. It explains how numerous issues in calculus could seem unrelated yet in fact have universal roots. Emphasizing old views, the textual content supplies scholars a glimpse into the advance of calculus and its rules from the age of Newton and Leibniz to the 20th century. approximately three hundred examples result in very important theorems in addition to aid scholars strengthen the required talents to heavily learn the theorems. Proofs also are awarded in an available strategy to students.

By strengthening abilities received via simple calculus, this textbook leads scholars towards gaining knowledge of calculus concepts. it is going to aid them reach their destiny mathematical or engineering studies.

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Example text

Let L = sup V (an ). We will show that L is an accumulation point (hence the largest one). Let ε > 0. Then L − ε is not an upper bound for V (an ), so there exists a ∈ V (an ) such that L− ε < a ≤ L. 2 Thus, |L − a| < ε/2. Since a is an accumulation point of {an }, there exists n ∈ N such that |an − a| < ε/2. Now, for that n, |L − an | ≤ |L − a| + |a − an | < ε ε + = ε. 2 2 We conclude that L is an accumulation point of {an }, and we will leave the existence of the smallest accumulation point as an exercise.

If A and B are non-empty bounded subsets of R and A ⊂ B, prove that sup A ≤ sup B. 10, determine whether it is bounded and find sup A and inf A if they exist. 5. A = {x : x2 < 3x}. 6. A = {x : 2x2 < x3 + x}. m 4n + : m, n ∈ N . 8. 7. A = {x : 4x2 > x3 + x}. 9. A = : m, n ∈ N . 10. A = : m, n ∈ N . 11. Let A be a non-empty subset of R with the property that sup A = inf A. Prove that the set A has precisely one point. 12. Let A be a non-empty subset of R and let f, g be functions defined on A. (a) Prove that sup{f (x) + g(x) : x ∈ A} ≤ sup{f (x) : x ∈ A} + sup{g(x) : x ∈ A}.

Now we can consider a very important sequence which is the central topic of this section. 2. The sequence an = 1+ 1 n n is convergent. Proof. First we will show that an is an increasing sequence. Instead of an+1 − an > 0, we will establish that an+1 /an > 1. ) We notice that n+1  n+1 1 1 1+ 1 +  1 an+1 n+1 n+1  1+ = . = n  an n 1 1  1+ 1+ n n 18 Advanced Calculus: Theory and Practice Further, 1 n+1 = 1 1+ n 1+ n+2 n+1 n+1 n = n2 + 2n 1 n(n + 2) = =1− 2 . (n + 1)2 n2 + 2n + 1 n + 2n + 1 Therefore, using Bernoulli’s Inequality,  n+1 1 1 +  1 1 n n+1   ≥ 1 − (n + 1) · 2 =1− = .

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