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

By John Srdjan Petrovic

ISBN-10: 1351381431

ISBN-13: 9781351381437

ISBN-10: 1466565640

ISBN-13: 9781466565647

ISBN-10: 2592642692

ISBN-13: 9782592642691

Suitable for a one- or two-semester path, Advanced Calculus: idea and Practice expands at the fabric lined in undemanding calculus and offers this fabric in a rigorous demeanour. The textual content improves scholars’ problem-solving and proof-writing abilities, familiarizes them with the old improvement of calculus options, and is helping them comprehend the connections between diverse topics.

The publication takes a motivating procedure that makes rules much less summary to scholars. It explains how a number of subject matters in calculus could appear unrelated yet actually have universal roots. Emphasizing historic 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 support scholars advance the mandatory talents to heavily research the theorems. Proofs also are provided in an available approach to students.

By strengthening abilities won via trouble-free calculus, this textbook leads scholars towards learning calculus options. it is going to support them reach their destiny mathematical or engineering studies.

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Additional resources for Advanced Calculus: Theory and Practice

Sample text

Let an , bn > 0 for all n ∈ N. Prove that lim sup(an bn ) ≤ lim sup an lim sup bn . Give an example to show that the inequality may be strict. Give an example to show that the assumption an , bn > 0 cannot be relaxed. 5. Let {an } be a sequence of positive numbers and let {bn } be a convergent sequence of positive numbers. Prove that lim sup(an bn ) = lim sup an lim bn . 6. Let {an } be an increasing sequence that has a bounded subsequence. Prove that the sequence {an } is convergent. 7. Let {an } be a sequence such that every subsequence {ank } contains a convergent subsequence {ankj } converging to L.

N+1 22 Advanced Calculus: Theory and Practice n 1 . Prove that lim an = e. k=1 k(k + 1)(k + 1)! 8. Prove that the sequence an = 1 + + + · · · + − ln n is increasing and bounded 2 3 n above. Conclude that it is convergent. 7. 8. 5772. 9. 6. 10. A sequence {an } is a geometric sequence if there exists q > 0 such that an+1 /an = q, n ∈ N. A sequence {bn } is an arithmetic sequence if there exists d > 0 such that an+1 −an = d, n ∈ N. If {an } and {bn } are such sequences, and if a1 = b1 > 0, a2 = b2 > 0, prove that an > bn for n ≥ 3.

N n! Solution. We will show first that n! > (n/3)n , using induction. When n = 1 we have 1 > 1/3, so suppose that it is true for n, and let us establish this inequality for n + 1. Using the hypothesis, (n + 1)! = (n + 1)n! > (n + 1) n 3 n = n+1 3 Therefore, n! > (n/3)n for all n ∈ N. Consequently, n+1 1 3 0< √ < →0 n n n! and lim an = 0. 3 1 + n1 n > n+1 3 n+1 . 6. Let lim an = a and bn = . Prove that the sequence {bn } n is convergent and that lim bn = a. Solution. Let ε > 0. Then, there exists N1 ∈ N, such that for any n ≥ N1 , we have |an − a| < ε .

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Advanced Calculus: Theory and Practice by John Srdjan Petrovic

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