By Steven G. Krantz
Do not get me unsuitable - Krantz is sweet yet this can be primarily child Rudin - with no the proofs - that's type 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
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. This is of theoretical interest, but can alsc5 be applied to determine the irrationality of e. 7 With AN as above. ,. N N! With some sharp theoretical work. 8 Euler's number e is irrational. For a reference, see [RUD) or [KRAl).
Sin2 j an L . j=1 J diverge. The details of these assertions are left to the reader. 3 Absolute and Conditional Convergence We tum next to the topic of absolute and conditional convergence. A series of real or complex constants is said to be absolutely convergent if converges. 5 If the se£ies L:~=I aj is absolutely convergent, then it is convergent. 1 A series L~=I a} is said to be conditionally convergent if converges. but it does not converge absolutely. L~=J a} We see that absolutely convergent series are convergent.
In the course of calculating. you will have produced elements of the sequence that are arbitrarily near to 0, and you will also have produced elements that are arbitrarily near to 1. Thus your calculations illustrate the proposition. 0 We conclude this brief consideration of lim sup and lim inf with a result that ties all the ideas together. 6 Let (a j} be a sequence of real numbers. m inf aj. 3 17 Some Special Sequences If (a jk I is any subsequence of the given sequence, then Moreover, there is a subsequence (a j, I such that and another sequence (a jm I such that lim ajm = m_oo p.
A Handbook of Real Variables: With Applications to Differential Equations and Fourier Analysis by Steven G. Krantz