S. Zaidman.'s Advanced calculus : an introduction to mathematical analysis PDF

By S. Zaidman.

ISBN-10: 9810227043

ISBN-13: 9789810227043

Ch. 1. Numbers --
ch. 2. Sequences of genuine numbers --
ch. three. endless numerical sequence --
ch. four. non-stop services --
ch. five. Derivatives --
ch. 6. Convex capabilities --
ch. 7. Metric areas --
ch. eight. Integration.

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Example text

Remark 2. If a n —► 0 and /Jn -» 00, the sequence ( a n • /3n) can converge to a finite limit or to +00. n = n, gives a n /3 n -► 0; an = £, /3 n = n 2 gives a n /3 n = n -► 00; «n = f , Pn = n gives an(3n = B -+ B). 3 Monotone Sequences Monotone sequences of real numbers have been introduced already, in connec­ tion with the "upper bound property". We previously proved that an increasing upper bounded sequence is a Cauchy sequence, hence it has a limit. We also used this result in order to prove that any nonempty set of real numbers which is upper bounded has a (unique) least upper bound.

The root test). Suppose an > 0 for all n. If lim sup n v/a^ CO CO l l < 1, then the series 53 an converges. If lim sup nyfa~^ > 1 £/ien t/ie series 53 a n diverges. Proof. If lim sup 7\fan~ = q < 1 we take q', q < q' < 1 and we find no € N such that y ^ < q' for n > no- Hence an < {q')n for n > no, and therefore CO 53 &n is convergent. l If lim sup ny/a^ = r > 1 we take r', 1 < r1 < r. 4 (II) (2). Take ni > 1, such that n\fa~^[ > r'. Then, take n 2 > ni, such that n \fa^ > r', continuing this way we find a subsequence (a,nk)kLi of (a n ) such 54 Advanced Calculus that aUk > r'nk.

It appears in many fundamental formulas of analysis. (i) First, the monotonicity relation: xn < #n+i> Vn £ N, can be established in a rather elementary way, by simple use of a strict form of Bernoulli's inequality. (see below) This inequality in fact implies that, for n > 2 'n2-l\ n = A n * 1 - —z \ n J Then consider the quotient ^ 1 , 1 > l - n - - jz = l - - = n n = (gf^r = ^ S ^ - ^ n-1 . • T h u s xn > *n-i for n = 2 , 3 , . . (Here we used the "strict" Bernoulli's inequality: If x > — 1 and x / 0 , then (1 + x)n > 1 + nx for all n > 2: again, it is true for n = 2; assume it true for n = m; thus (1 + x ) m > 1 -f rare.

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