By Steven G. Krantz

ISBN-10: 376434329X

ISBN-13: 9783764343293

Do not get me unsuitable - Krantz is sweet yet this is often basically child Rudin - with out the proofs - that's kind of like a bar with no beer.

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We first need a definition. 2 Let L~=I c} be a given series. ,1 be a sequence in which every positive integer occurs once and only once (but not necessarily in the usual order). Then the series k LC Pi }=I is said to be a rearrangement of the given series. 7 Weierstrass, Riemann If the series :L'=t aj of real numbers is absolutely convergent, then of course it is convergent; let the sum be e. Then every rearrangement of the series converges also tot. If the series :L'=t bj is conditionally convergent and if {3 is any real number or ±oo, then there is a rearrangement of the series such that its sequence ofpartial sums converges to {3.

We already know that the series diverges. However, the expression sin j changes sign in a rather sporadic fashion. 1) converges because of cancellation of the summands. We take aJ = sinj and bJ = I/j. Abel's test will apply if we can verify that the partial sums AN of the aj's are bounded. To see this, we use a trick: Observe that cos(j + 1/2) = cos j . cos(l/2) - sin j . sin(l/2) and cos(j - 1/2) = cos j . cos(l/2) + sin j . sin(1/2). Subtracting these equations and solving for sin j yields that .

7). Continue to bisect and choose, at each stage selecting a subinterval that contains infinitely many elements of the sequence and an element ait that is further along in the sequence. In this manner we obtain the desired subsequence. , the limit point). 1 Introduction Compact sets are sets (usually infinite) which share many of the most important properties of finite sets. They play an important role in real analysis. 4 A set S C lR is called compact if every sequence in S has a subsequence that converges to an element of s.

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

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