In this section, we introduce sequences and define what it means for a sequence to converge or diverge. 5.1.3 Determine the convergence or divergence of a given sequence. 5.1.2 Calculate the limit of a sequence if it exists. 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. Learning Objectives 5.1.1 Find the formula for the general term of a sequence. Find the first ten terms of p n p n and compare the values to π. 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. Therefore, being bounded is a necessary condition for a sequence to converge. For example, consider the following four sequences and their different behaviors as n → ∞ n → ∞ (see Figure 5.3): Not every sequence has this behavior: those that do. Since a sequence is a function defined on the positive integers, it makes sense to discuss the limit of the terms as n → ∞. The limit of a sequence is the value the sequence approaches as the number of terms goes to infinity. Limit of a SequenceĪ fundamental question that arises regarding infinite sequences is the behavior of the terms as n n gets larger. Find an explicit formula for the sequence defined recursively such that a 1 = −4 a 1 = −4 and a n = a n − 1 + 6.
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