Download A Course in Calculus and Real Analysis (Undergraduate Texts by Sudhir R. Ghorpade, Balmohan V. Limaye PDF

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By Sudhir R. Ghorpade, Balmohan V. Limaye

This booklet presents a self-contained and rigorous advent to calculus of features of 1 variable. The presentation and sequencing of themes emphasizes the structural improvement of calculus. while, due value is given to computational concepts and functions. The authors have strived to make a contrast among the intrinsic definition of a geometrical thought and its analytic characterization. in the course of the ebook, the authors spotlight the truth that calculus presents an organization starting place to a number of options and effects which are mostly encountered in highschool and authorized on religion. for instance, you can find right here an evidence of the classical consequence that the ratio of the circumference of a circle to its diameter is identical for all circles. additionally, this e-book is helping scholars get a transparent realizing of the concept that of an attitude and the definitions of the logarithmic, exponential and trigonometric features including an explanation of the truth that those should not algebraic features. a few issues which can were inadequately lined in calculus classes and glossed over in genuine research classes are taken care of right here in substantial element. As such, this booklet presents a unified exposition of calculus and actual analysis.

The purely necessities for interpreting this ebook are themes which are often coated in highschool; despite the fact that, the reader is predicted to own a few mathematical adulthood and a capability to appreciate and take pleasure in proofs. This ebook can be utilized as a textbook for a significant undergraduate path in calculus, whereas elements of the ebook can be utilized for complicated undergraduate and graduate classes in actual research. every one bankruptcy comprises a number of examples and a wide number of workouts, in addition to "Notes and Comments" describing salient good points of the exposition, comparable advancements and references to correct literature.

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Thus if we replace an1 , . . , ank by bn1 , . . , bnk respectively, then the altered sequence converges if and only if the original sequence converges. With this in view, we may sometimes regard (1/an ) as a sequence if we know that all except finitely many an ’s are nonzero. Next, if an → a, then the inequality | |an | − |a| | ≤ |an − a|, n ∈ N, shows that |an | → |a|. The converse is not true as can be seen by considering an = (−1)n for n ∈ N. However, if |an | → 0, then clearly an → 0. 2. (i) A convergent sequence has a unique limit.

Given any m, n ∈ Z, show that an LCM of m and n exists and is unique; it is denoted by LCM(m, n). Also show that if m and n are nonnegative integers and we let d = GCD(m, n) and ℓ = LCM(m, n), then dℓ = mn. 42. If m, n, n′ ∈ Z are such that m and n are relatively prime and m | nn′ , then show that m | n′ . Deduce that if p is a prime (which means that p is an integer > 1 and the only positive integers that divide p are 1 and p) and if p divides a product of two integers, then it divides one of them.

A variety of proofs of the Fundamental Theorem of Algbra, which implies the Real Fundamental Theorem of Algebra stated in this chapter, can be found in the book of Fine and Rosenberger [25]. For real-valued functions defined on intervals, we have discussed a number of geometric properties such as monotonicity, convexity, local extrema, and the Intermediate Value Property. Typically, these appear in calculus books in conjunction with the notions of differentiability and continuity. The reason to include these in the first chapter is to stress the fact that these are geometric notions and should not be confused with various criteria one has, involving differentiability or continuity, to check them.

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