Download A Treatise On The Differential Calculus with numerous by Isaac Todhunter PDF

By Isaac Todhunter

This Elibron Classics booklet is a facsimile reprint of a 1864 variation via Macmillan and Co., Cambridge and London.

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27 Let {aj } be a increasing sequence of real numbers. Then the sequence has a limit—either a finite number or +∞. Let {bj } be a decreasing sequence of real numbers. Then the sequence has a limit—either a finite number or −∞. 1 Some books say “converging to infinity,” but this terminology can be confusing. 28 If S is a set of real numbers which is not bounded above, we say that its supremum (or least upper bound) is +∞. If T is a set of real numbers which is not bounded below then we say that its infimum (or greatest lower bound) is −∞.

If j is any natural number then either 1 ≤ j ≤ N (in which case |aj | ≤ Q) or else j > N (in which case |aj | ≤ P ). Set M = max{P, Q}. Then |aj | ≤ M for all j, as desired. So the sequence is bounded. The next proposition records some elementary properties of limits of sequences. 1. 6 Let {aj } be a sequence of real or complex numbers with limit α and {bj } be a sequence of real or complex numbers with limit β. Then we have: (1) If c is a constant then the sequence {c · aj } converges to c · α; (2) The sequence {aj + bj } converges to α + β; (3) The sequence aj · bj converges to α · β; (4) If bj = 0 for all j and β = 0 then the sequence aj /bj converges to α/β.

If α = 0 then j α is a constant sequence, identically equal to 1. If α < 0 then j α = 1/j −α . The denominator of this last expression tends to +∞ hence the sequence j α tends to 0. 30 CHAPTER 2. 38 The sequence {j 1/j } converges to 1. In fact, consider the expressions αj = j 1/j − 1 > 0. We have that j = (αj + 1)j ≥ j(j − 1) (αj )2 , 2 (the latter being just one term from the binomial expansion). Thus 0 < αj ≤ 2/(j − 1) as long as j ≥ 2. It follows that αj → 0 or j 1/j → 1. 39 Let α be a positive real number.

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