![]() 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. ![]() Answer: First, we prove by induction that 0 < an 2 for all n. Find the first ten terms of p n p n and compare the values to π. 100 (1 rating) Transcribed image text: Q5 (15 points) Prove that the following recursive sequence converges and find the limit: Q1 1 and an+1 5an 3+ an. Determine whether the sequence (an) converges or diverges. 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): Since a sequence is a function defined on the positive integers, it makes sense to discuss the limit of the terms as n → ∞. ![]() induction to prove a formula for the sum of the first n integers. Limit of a SequenceĪ fundamental question that arises regarding infinite sequences is the behavior of the terms as n n gets larger. In general, mathematical induction is a method for proving. Solution 1 You need to investigate first whether a n is convergent or not. Find an explicit formula for the sequence defined recursively such that a 1 = −4 a 1 = −4 and a n = a n − 1 + 6. A necessary and sufficient condition is found for a linear recursive sequence to be convergent, no matter what initial values are given. ![]()
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