Download e-book for kindle: A Handbook of Real Variables: With Applications to by Steven G. Krantz

By Steven G. Krantz

ISBN-10: 376434329X

ISBN-13: 9783764343293

Do not get me incorrect - Krantz is sweet yet this is often basically child Rudin - with out the proofs - that is kind of like a bar with out beer.

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Download e-book for iPad: A Handbook of Real Variables: With Applications to by Steven G. Krantz

Do not get me improper - Krantz is nice yet this can be primarily child Rudin - with out the proofs - that's kind of like a bar with no beer.

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Additional info for A Handbook of Real Variables: With Applications to Differential Equations and Fourier Analysis

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J- = where j! = j . (j - 1) . 1 for j ~ I and O! I, is convergent (by the Ratio Test, for instance). Its sum is denoted by the symbol e in honor of the Swiss mathematician Leonard Euler, who first studied it. Like the number 11:, to be considered later in this book. the number e is one that arises repeatedly in a variety of contexts in mathematics. It has many special properties. 6 The limit lim n-+OO (1+-nI)" exists and equals e. 19. I! of the series The next result tells us how rapidly the partial sums AN = defining e converge to e.

Observe that. whereas any open set is the union of open intervals, the existence of the Cantor set shows us that there is no such structure theorem for closed sets. In fact, closed intervals are atypically simple examples of closed sets. 1 Connectivity Let S be a set of real numbers. We say that S is disconnected if it is possible to find a pair of open sets U and V such that un S i= 0, V n S i= 0, (U and n S) n (V n S) = 0, s=(uns)u(vns). If no such U and V exist, then we call S connected. 9. 9 Example 4.

We can perform the division to see that IC~;II = j:I' The lim sup of the last expression is O. By the Ratio Test, the series converges. Notice that in this example. while the Root Test applies in principle. it would be difficult to use in practice. 11 We apply the Root Test to the series 00 ·2 L~ 2J j=l Observe that ·2 J CJ =-. 2 Elementary Convergence Tests hence As j .... 00, we see that lim sup ICjl II' J 1 = -2 < 1. 1-+00 By the Root Test, the series converges. 6 Root and Ratio Tests for Divergence It is natural to ask whether the Ratio and Root Tests can detect divergence.

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A Handbook of Real Variables: With Applications to Differential Equations and Fourier Analysis by Steven G. Krantz


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