įor the following two exercises, assume that you have access to a computer program or Internet source that can generate a list of zeros and ones of any desired length. Learn how to convert between recursive and explicit formulas of arithmetic sequences. induction to prove a formula for the sum of the first n integers. 1, 4 The familiar theorem of classical analysis that a monotone bounded sequence is convergent does not hold in recursive analysis. Find the first ten terms of p n p n and compare the values to π. In general, mathematical induction is a method for proving. To find an approximation for π, π, set a 0 = 2 + 1, a 0 = 2 + 1, a 1 = 2 + a 0, a 1 = 2 + a 0, and, in general, a n + 1 = 2 + a n. In particular, squences that go off to plus or minus infinity. Now we will look at some specific ways that sequences can diverge. Therefore, being bounded is a necessary condition for a sequence to converge. (last updated: 3:41:39 PM, October 06, 2020) 2.4 - Monotone sequences You have now seen a variety of sequence theorems and are familiar with sequences converging to finite real numbers. For example, consider the following four sequences and their different behaviors as n → ∞ n → ∞ (see Figure 5.3): Since a sequence is a function defined on the positive integers, it makes sense to discuss the limit of the terms as n → ∞. Example 4 Consider a sequence de ned recursively, a 1 p 2 and a n 2 + p a n 1 for n 2 3 :. Theorem 9 (Monotone Convergence) A monotone sequence is convergent if and only if it is bounded. Moreover, a monotone sequence converges only when it is bounded. Limit of a SequenceĪ fundamental question that arises regarding infinite sequences is the behavior of the terms as n n gets larger. Here, we prove that if a bounded sequence is monotone, then it is convergent. Find an explicit formula for the sequence defined recursively such that a 1 = −4 a 1 = −4 and a n = a n − 1 + 6. Let us consider a recursive sequence of x x + sinx and the initial value x.
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