By John Srdjan Petrovic

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

The e-book takes a motivating method that makes principles much less summary to scholars. It explains how numerous issues in calculus could seem unrelated yet actually have universal roots. Emphasizing historic views, the textual content supplies scholars a glimpse into the improvement of calculus and its rules from the age of Newton and Leibniz to the 20 th century. approximately three hundred examples bring about vital theorems in addition to aid scholars advance the mandatory talents to heavily research the theorems. Proofs also are provided in an available option to students.

By strengthening abilities received via undemanding calculus, this textbook leads scholars towards getting to know calculus thoughts. it is going to aid them reach their destiny mathematical or engineering studies.

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**Extra resources for Advanced Calculus: Theory and Practice**

**Example text**

Lim{(2 + 3) }, where {x} = x − ⌊x⌋. 4. lim nq n , if |q| < 1. 5. lim(1 + 3n)1/n . 6. lim 2−1/ n . 7. Given x ≥ 1, show that lim(2 n x − 1)n = x2 . 8. Let {an } be a sequence of positive numbers. Prove that lim sup a1 + an+1 an n ≥ e. 9. Suppose that the terms of the sequence {an } satisfy the inequalities 0 ≤ an+m ≤ an + am . Prove that the sequence {an /n} converges. 6). In this chapter, our goal is to prove these theorems. When Cauchy did that in Cours d’analyse he took some properties of real numbers as self-evident.

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}. (b) Prove that inf{f (x) + g(x) : x ∈ A} ≥ inf{f (x) : x ∈ A} + inf{g(x) : x ∈ A}. (c) Give examples to show that each of the inequalities in (a) and (b) can be strict. 7). Now that we have the Completeness Axiom we can do not only that, but we can also derive several equally important results.

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. Use this result to derive the Bernoulli’s Inequality. 4 we first encountered the situation where our main goal was to prove that a sequence converges.