This is clear since x n 1 g ( x n) 3 4 x n. However, if we know that a recursive sequence converges, it is usually easy to show what it converges to using the following observation: lim an-1 lim an (. Since convergence depends only on what happens as \(n\) gets large, adding a few terms at the beginning can't turn a convergent sequence into a divergent one. There is a theorem that says for any recursion x n 1 g ( x n), convergence is guaranteed whenever g ( x) < 1. (c) Use the monotonic convergence theorem to conclude this sequence has a limit and then find that limit. But starting with the term \(3/4\) it is increasing, so the theorem tells us that the sequence \(3/4, 7/8, 15/16, 31/32,\ldots\) converges. (b) Prove that this sequence is always increasing. Power Series and Polynomial Approximationįrequently these formulas will make sense if thought of either as functions with domain \(\mathbb\) \(31/32,\ldots\) is not increasing, because among the first few terms it is not. ![]() First Order Linear Differential Equations.Triple Integrals: Volume and Average Value.Double Integrals: Volume and Average Value.Monotone Sequence Theorem: (sn) is increasing and bounded above, then (sn) converges. Partial Fraction Method for Rational Functions Further, PMI is a main tool in proving the correctness of recursive. The following theorem gives averyelegant criterion for a sequence toconverge, and explains why monotonicity is so important.Open Educational Resources (OER) Support: Corrections and Suggestions.
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