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Help in proving sequence by induction

  1. Sep 1, 2007 #1

    rock.freak667

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    1. The problem statement, all variables and given/known data
    The sequence [tex]u_1,u_2,u_3,...[/tex] is such that [tex]u_1=1[/tex] and [tex]u_{n+1}=-1+{\sqrt{u_n+7}}[/tex]

    a) Prove by induction that [tex]u_n<2 for all n\geq1[/tex]
    b) show that if [tex]u_n=2-\epsilon[/tex], where [tex]\epsilon[/tex] is small, then [tex]u_{n+1}\approx 2-\frac{1}{6}\epsilon[/tex]



    2. Relevant equations

    3. The attempt at a solution
    [tex]u_{n+1}=-1+sqrt{u_n+7}[/tex]

    [tex]\Rightarrow u_n=(u_{n+1}+1)^2-7[/tex]

    Assume statement is true for all [tex]k\geq1[/tex]
    then [tex]u_k<2[/tex]
    [tex]\Rightarrow (u_{k+1}+1)^2-7<2[/tex]


    [tex] (u_{k+1}+1)^2-(3)^2<0[/tex]

    [tex]((u_{k+1}+1)-3)((u_{k+1}+1)-3)<0[/tex]

    [tex](u_{k+1}+1)-3>0[/tex] AND [tex](u_{k+1}+1)-3<0[/tex]
    [tex]
    u_{k+1}+1>3

    u_{k+1}>2
    [/tex]

    Thus [tex]u_{n+1}>2[/tex] is true

    [tex]
    (u_{k+1}+1)-3<0

    u_{k+1}+1<-3

    u_{k+1}<-2
    [/tex]
    does this affect anything in my proof?

    I didn't bother to substitute the values of [tex]u_1[/tex] and [tex]u_2[/tex] and so forth as i have already done it and it is so for all [tex]n\geq1[/tex]

    but I do not know how to do part b)
     
    Last edited by a moderator: Sep 2, 2007
  2. jcsd
  3. Sep 1, 2007 #2

    learningphysics

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    Didn't you mean assume it's true for [tex]n\leq{k}[/tex] ? Actually assuming it is true for n=k is sufficient. But either way is fine.

    Good so far... but at this point I'd take the 3^2 to the other side... if x^2<a^2 where a>0, then x<a and x>-a... and then you can use the x<a part to finish...
     
  4. Sep 1, 2007 #3

    learningphysics

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    For part b, use the taylor series expansion in terms of [tex]\epsilon[/tex]... you only need the first two terms.
     
  5. Sep 1, 2007 #4

    rock.freak667

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    So then expand [tex]\sqrt{u_n+7}[/tex] using taylor series and it should work out then?
     
  6. Sep 1, 2007 #5

    learningphysics

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    You should substitute in [tex]2-\epsilon[/tex] for [tex]u_n[/tex] in the expression for [tex]u_{n+1}[/tex]... then get the taylor series of [tex]u_{n+1}[/tex] in terms of epsilon...
     
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