![]() ![]() Given a sequence a n : n ∈ N, one can associate another sequence s n : n ∈ N to it, where s n = Σ a k : k ≤ n denotes the sum of the first n + 1 terms of a n : n ∈ N. A sequence b n : n ∈ N tends to –∞ if, for every r n 0. A sequence a n : n ∈ N tends to +∞ if, for every r > 0, there exists an n 0 = n 0 ( r ) such that a n > r whenever n > n 0. ![]() Sometimes it is convenient to treat certain divergent sequences as if they were converging to infinity. The sequence a n : n ∈ N, where a n = ( n ! ) 1 / n for n > 0, is unbounded, and thus divergent. The sequence a n : n ∈ N, where a n = 1 for all even n, and a n = - 1 for all odd natural n, is divergent. Example 4.Īny unbounded sequence is divergent in particular, the unbounded sequence b n : n ∈ N, where b n = 1 + 1 / 2 + … + 1 / n for n > 0. In some sense, most real sequences are divergent. If there exists no limit of a sequence a n : n ∈ N, then the sequence is divergent. The sequence a n : n ∈ N, where a n = 1 + 1 / 2 + … + 1 / n - ln ( n ) for n > 0, converges to Euler’s constant, C = 0.57721…. The sequence a n : n ∈ N, where a n = 1 + 1 / n n for n > 0, converges to Napier’s constant, e = 2.71828…. The sequence 1, 2 1 / 2, 3 1 / 3, …, n 1 / n, … converges and its limit is equal to 1. The convergence of a sequence of real numbers depends only on its tail. Then a n : n ∈ N converges if and only if b n : n ∈ N converges and, in this case, so does the equality Suppose that, for two sequences a n : n ∈ N and b n : n ∈ N, two natural numbers m and n can be found such that a m + i = b n + i whenever i ∈ N. ![]() If two sequences a n : n ∈ N and b n : n ∈ N converge to a and b respectively, then, for any real r and t, the sequence r a n + t b n : n ∈ N converges to r a + t b. If a sequence of real numbers is monotone and bounded, then it necessarily has a limit. Many other conditions for convergence are more convenient in practice. The Cauchy criterion reflects the completeness property of the real numbers, from which necessary and sufficient condition for the existence of a limit follow almost at once. 3Īmong the glories of mathematical analysis, no one beyond Cauchy has ever found these ideas easy, and no one beyond Leonhard Euler has ever found them obvious. The classical criterion-by Cauchy again, as it happens again-affirms that a n : n ∈ N is convergent if, and only if, for any ε > 0 there exists a natural number n 0 = n 0 ( ε ) such that a n - a m n 0 and m > n 0. The definition of a limit having been given-by Augustin-Louis Cauchy, as it happens-mathematicians wished naturally to know whether and under what conditions such limits existed. If this is the case, the sequence converges (or tends) to a as n tends to infinity: a = l i m a n : n ∈ N. A real number a is a limit of a sequence a n : n ∈ N if, for any ε > 0, there exists a natural number n 0 = n 0 ( ε ) such that a n - a n 0. The terms a n of such sequences are often generated by a fixed rule, or by applying a specific algorithm to each n. ![]() Thus a n : n ∈ N or more explicitly, if less compactly An infinite sequence is an arbitrary function whose domain coincides with the set ℕ = of natural numbers, or with some of its infinite subsets, and whose range is contained in the real line R. For the moment, I will consider the real numbers and only the real numbers. M athematical analysis begins by considering infinite sequences of real numbers and by defining their limit. 1 Having made outstanding contributions to the subject as the Devil’s plaything, he knew what he was talking about. Niels Abel called divergent series the Devil’s invention. Surely this sequence of numbers is simply getting bigger and bigger and beyond this, it is not converging to anything. What, for example is the sum of 1 + 2 + 3 + … + n + …? What makes the interplay intriguing is the emergence from the most elementary considerations of results that are paradoxical, or, at least, utterly counterintuitive. C lassical mathematical analysis involves an intriguing interplay between finite and infinite collections and between discrete and continuous structures. ![]()
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