Download e-book for iPad: Advanced Calculus: Theory and Practice by John Srdjan Petrovic

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 direction, Advanced Calculus: thought and Practice expands at the fabric lined in basic calculus and offers this fabric in a rigorous demeanour. The textual content improves scholars’ problem-solving and proof-writing talents, familiarizes them with the ancient improvement of calculus suggestions, and is helping them comprehend the connections between assorted topics.

The ebook takes a motivating procedure that makes rules much less summary to scholars. It explains how quite a few themes in calculus could appear unrelated yet in fact have universal roots. Emphasizing ancient views, the textual content supplies scholars a glimpse into the improvement of calculus and its principles from the age of Newton and Leibniz to the 20th century. approximately three hundred examples bring about very important theorems in addition to support scholars improve the required abilities to heavily research the theorems. Proofs also are offered in an obtainable solution to students.

By strengthening abilities won via simple calculus, this textbook leads scholars towards gaining knowledge of calculus recommendations. it's going to support them reach their destiny mathematical or engineering studies.

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Extra info for Advanced Calculus: Theory and Practice

Example text

Since qa < p < qb, we obtain that√ a <√ r < b. In order to construct ρ, we consider the interval ( b2 , a2 ). Once again, the Archimedean Property yields a positive integer p such that √ √ 2 2 > 1. p − a b √ Therefore, there exists a positive integer q between p a2 and p √ √ 2 2 p

6 that the sequence an = (−1)n is divergent. Nevertheless, we have a perfect understanding of its asymptotic behavior (meaning: as n → ∞). When n is an even number, say n = 2k with k ∈ N, then an = 1. On the other hand, when n is an odd number, say n = 2k − 1 with k ∈ N, then an = −1. In order to describe a situation like this we will develop some terminology first. Notice that the sequence a1 , a3 , a5 , . . is convergent. This is not the whole sequence {an }. 1. Let {an } be a sequence, and let n1 < n2 < n3 < .

2 2 2 Solution. If m ≥ n then |am − an | = ≤ ≤ = < sin m sin(n + 1) sin(n + 2) + + ···+ m 2n+1 2n+2 2 sin(n + 1) sin(n + 2) sin m + + ···+ 2n+1 2n+2 2m 1 1 1 + n+2 + · · · + m n+1 2 2 2 1 1 − 1/2m−n · 2n+1 1 − 1/2 1 1 1 · = n. 9) Given ε > 0 we will require that 1/2N < ε. This leads to 2N > 1/ε. By taking the natural logarithm of both sides, we have that N ln 2 > ln(1/ε), hence N > ln(1/ε)/ ln 2. In other words, ln 1ε 1 < ε. 10) if and only if N> ln 2 2N Proof. Let ε > 0 and let N = max{⌊ln(1/ε)/ ln 2⌋ + 1, 1}.

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Advanced Calculus: Theory and Practice by John Srdjan Petrovic

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