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13.6 Limits of Sequences

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We have looked at sequences, writing them out, summing them, etc. But, now let’s examine what they “go to” as n gets larger and larger without bound. Consider What are the terms going to? Pick a bigger number. Looks like they get close to 0. So, we say *Note: There is no term that IS 0, it just gets SUPER CLOSE to 0!

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To find the limit of a sequence, we usually have to algebraically manipulate it. Ex 1) Find the limit of a sequence as n increases without bound. a) b) goes to 0

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If a sequence gets closer to a number, L, as n increases without bound, it is said to converge and it is a convergent sequence. If it does not converge, the sequence is said to diverge. *Note: If it does diverge, it can do so by several ways – getting larger & larger or by oscillating. Ex 2) Find the first 5 terms and decide if the sequence converges or diverges. a) b) getting larger diverges diverges 1.5 1.75 2.3 2.8

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Ex 3) Determine whether these geometric sequences converge or diverge. a) b) c) d) diverge diverge converge converge Can you generalize these geometric sequences & make a rule for what will converge & what will diverge? Try On Your Own! Ex 4) Characterize each sequence as convergent or divergent. If it converges, give the limit. 0 convergent divergent n 00 1 0

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Homework #1306 Pg 718 #1–16 all, 18, 20, 22–27 all, 30, 32, 34–36

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