This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 lets extend our knowledge of Mathematica a little instead. New commands are If, OddQ . There is also a command EvenQ , but we wont 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, lets 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 isnt 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....
View
Full
Document
 Spring '08
 Keppelmann
 Calculus

Click to edit the document details