By Joseph R. Lee

ISBN-10: 0124407501

ISBN-13: 9780124407503

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2) Are the real numbers thus obtained themselves complete? 1 Completeness of the Real Number System 43 A detailed discussion of these questions would require considerable digression, and the student is referred to the work of Landau. + For our purposes, we shall assume the familiar algebraic properties of real numbers, a completeness axiom, and also the properties of the order relations " < " and " < " . * Among the last is the property that between any two numbers (rational or irrational) there is both a rational and an irrational number (and therefore an infinite number of each kind).

Then ΣΓ=ι ( — 1 ) *+1α* converges. 2n+l = «1 — («2 — dz) — ( α 4 — α 5 ) — · · · — ( α 2 η — «2n+l) . Then Ain is monotone nondecreasing and A2n+i is monotone nonincreasing. 4; and since lim^«, a2w+i = 0, these limits must be the same, say, A. Then, lim^«, An = A. 8 COROLLARY With the hypotheses stated above, A = An + Rn with limn-oo An = A. Then | Än | < αη+ι· Proof let If n is odd, let n = 2Jfc - 1. Then 0 < A2k-i — A < A2k-i — A2k = a2k. If n is even, let n = 2k. Then 0 < A — A2k < A2k+1 — A2k = a2k+i.

9. · ¿4«(n+l) Spaces and Infinite 00 1/2 · y, {nx)n 10. 11. Series 1 Σ—— · 00 (ηχ)"-1 n Sequence Λ Μ ^ On n=l * 00 /y»n ¿Í (l - *)"' CO I Σ—-— tí(l+x)00 7. Σ n2e~n:c. 12. n=l 13. 00 n=0 In Problems 11 and 12, find the sum. 2) which requires, among other things, that if x± Ç lp and x2 Ç lP) then #i + ar2 (: Zp. If p = 1, this means that the sum of two absolutely convergent series is absolutely convergent. We shall see (Problem 1) that the condition of absolute convergence can be relaxed and that the sum of two (absolutely or conditionally) convergent series converges, provided the order of addition is not disturbed.

### Advanced calculus with linear analysis by Joseph R. Lee

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