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

By John Srdjan Petrovic

Sequences and Their Limits Computing the LimitsDefinition of the restrict homes of Limits Monotone Sequences The quantity e Cauchy Sequences restrict better and restrict Inferior Computing the Limits-Part II genuine Numbers The Axioms of the Set R results of the Completeness Axiom Bolzano-Weierstrass Theorem a few options approximately RContinuity Computing Limits of services A assessment of capabilities non-stop features: A Read more...

summary: Sequences and Their Limits Computing the LimitsDefinition of the restrict homes of Limits Monotone Sequences The quantity e Cauchy Sequences restrict more desirable and restrict Inferior Computing the Limits-Part II actual Numbers The Axioms of the Set R effects of the Completeness Axiom Bolzano-Weierstrass Theorem a few ideas approximately RContinuity Computing Limits of capabilities A assessment of services non-stop features: a geometrical perspective Limits of capabilities different Limits homes of continuing features The Continuity of undemanding capabilities Uniform Continuity homes of continuing capabilities

Show description

Read Online or Download Advanced Calculus : Theory and Practice PDF

Best functional analysis books

Elliptic theory and noncommutative geometry

The publication bargains with nonlocal elliptic differential operators. those are operators whose coefficients contain shifts generated by means of diffeomorphisms of the manifold on which the operators are outlined. the most objective of the learn is to narrate analytical invariants (in specific, the index) of such operators to topological invariants of the manifold itself.

Measure Theory and Integration

Techniques integration through degree, instead of degree through integration.

Weighted inequalities in Lorentz and Orlicz spaces

This set of chosen papers of Klingenberg covers a number of the vital mathematical facets of Riemannian geometry, closed geodesics, geometric algebra, classical differential geometry and foundations of geometry of Klingenberg. His contributions to Riemannian geometry have been major within the huge, in addition to establishing a brand new period in worldwide Riemannian geometry.

Additional resources for Advanced Calculus : Theory and Practice

Example text

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.

In fact, it is also bounded above by 9/5. Indeed, if this were not true, there would be a ∈ A such that a > 9/5. 24 > 2, contradicting the assumption that a2 < 2. Can we do better than 9/5? It is easy to verify that 8/5 is also an upper bound of A. Is there a least upper bound? The answer depends once again on the set in which we operate. If we allow real numbers, then √ the answer is in the affirmative, and we can even pinpoint the least upper bound: it is 2. ) Clearly, if we restrict ourselves to the set of rational numbers, then the answer is in the negative.

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. Use this result to derive the Bernoulli’s Inequality.

Download PDF sample

Rated 4.34 of 5 – based on 8 votes