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Unformatted text preview: Calculus II Homework: Sequences Page 1 1) (11.1.48) Find the first 40 terms of the sequence defined by a n +1 = a n 2 a n even 3 a n + 1 a n odd and a 1 = 11. Do the same if a 1 = 25. Make a conjecture about this type of sequence. 2) (11.1.49) For what values of r is the sequence { nr n } convergent? 3) (11.1.59) Find the limit of the sequence { √ 2 , p 2 √ 2 , q 2 p 2 √ 2 , . . . } . 4) (11.1.60) A sequence is given by a 1 = √ 2, a n +1 = √ 2 + a n . (a) By induction or otherwise, show { a n } is increasing and bounded above by 3. Show the sequence is convergent. (b) Find lim n →∞ a n . Solutions 1) We could work this out by hand, but let’s extend our knowledge of Mathematica a little instead. New commands are If, OddQ . There is also a command EvenQ , but we won’t need it for this problem. The original sequence was given as a 1 = 11 , a n +1 = a n 2 a n even 3 a n + 1 a n odd which has n = 1 , 2 , 3 . . . . However, to input it into Mathematica we prefer the following a 1 = 11 , a n = a n 1 2 a n 1 even 3 a n 1 + 1 a n 1 odd which has n = 2 , 3 , 4 . . . . Here are the Mathematica commands to define the sequence: a[1] := 11 a[n_] := a[n] = If[OddQ[a[n  1]], 3a[n  1] + 1, a[n  1]/2] I treated the sequence with different starting value as a totally new sequence, and defined it as b[1] := 25 b[n_] := b[n] = If[OddQ[b[n  1]], 3b[n  1] + 1, b[n  1]/2] Calculus II Homework: Sequences Page 2 The sequences are found to be: { a n } = { 11 , 34 , 17 , 52 , 26 , 13 , 40 , 20 , 10 , 5 , 16 , 8 , 4 , 2 , 1 , 4 , 2 , 1 , 4 , 2 , 1 , 4 , 2 , 1 , 4 , 2 , 1 , 4 , 2 , 1 , 4 , 2 , 1 , 4 , 2 , 1 , 4 , 2 , 1 , 4 } { b n } = { 25 , 76 , 38 , 19 , 58 , 29 , 88 , 44 , 22 , 11 , 34 , 17 , 52 , 26 , 13 , 40 , 20 , 10 , 5 , 16 , 8 , 4 , 2 , 1 , 4 , 2 , 1 , 4 , 2 , 1 , 4 , 2 , 1 , 4 , 2 , 1 , 4 , 2 , 1 , 4 } To make the connection easier to see, let’s write a small table. n a n b n 18 1 10 19 4 5 20 2 16 21 1 8 22 4 4 23 2 2 24 1 1 25 4 4 26 2 2 The two sequences become the same after the n = 22! This isn’t surprising, since the only thing that changed was the initial starting point...but then again, it is surprising, since a different starting point you might think would lead to different values later on. The sequence is oscillating, so it does not converge.would lead to different values later on....
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 Spring '08
 Keppelmann
 Calculus, Natural number

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