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Difficult sequence problem

  1. Nov 14, 2008 #1
    1. The problem statement, all variables and given/known data

    Use the triangle inequality (many times) and the formula for the partial sums of a geometric series to show that for m>n

    |s_m - s_n| <= r^(n-1)*(1/1-r)|s_2 - s_1|



    2. Relevant equations

    geometric series s = 1/1-r = 1 + r + r^2 + r^3...



    3. The attempt at a solution

    my first step was to multiply the terms inside the absolute to get 1/(1-r)

    |s_m - s_n| <= r^(n-1)*|(s_2/1-r) - (s_1/1-r)|

    next I expanded the geo. series as follows

    |s_m - s_n| <= r^(n-1) *

    |(s_2 + (s_2)*r + (s_2)*r^2 +...+ (s_2)*r^(m-1)) - |(s_1 + (s_1)*r + (s_1)*r^2 +...+(s_

    1)*r^(m-1))|

    then I applied the triangle inequality... many times

    r^(n-1)*|(s_2 + (s_2)*r + (s_2)*r^2 +...+ (s_2)*r^(m-1)) - |(s_1 + (s_1)*r + (s_1)*r^2 +...+(s_1)*r^(m-1))|

    <=

    r^(n-1)*|(s_2 - s_1)| + r^(n)*|(s_2 - s_1)| + r^(n+1)*|(s_2 - s_1)| + ... + r^(m+n-2)*|(s_2 - s_1)|

    is this close to the right path or did I make a mistake? thanks!
     
  2. jcsd
  3. Nov 14, 2008 #2

    HallsofIvy

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    An even more relevant equation is that [itex]s_m= 1+ r+ r^2+ ...+ r^m[/itex]

    But you don't know this is true. This is what you WANT to prove.

    I think you are going on the right path- but backwards!

    I would do this: [itex]|s_m- s_n|\le |s_m- s_(m-1)|+ |s_(m-1)+ s_n|\le |s_m- s_(m-1)|+ |s_(m-1)- s_(m-2)|+ |s_(m-2)- s_n| etc. until you have steps of 1 from s_m to s_n. Then use the fact that s_(k+1)- s_k= r^k so you have a sum of r^k from r^n to r^m. Then factor out r^n.
     
  4. Nov 15, 2008 #3
    [itex]|s_m- s_n|\le |s_m- s_(m-1)|+ |s_(m-1)+ s_n|\le |s_m- s_(m-1)|+ |s_(m-1)- s_(m-2)|+ |s_(m-2)- s_n| etc. until you have steps of 1 from s_m to s_n. Then use the fact that s_(k+1)- s_k= r^k so you have a sum of r^k from r^n to r^m. Then factor out r^n. [/itex]
     
  5. Nov 15, 2008 #4
    thank you for your time. When i use the fact that

    [itex]s_(k+1) + s_k = r^k[/itex]

    [itex]|s_m - s_n|<=r^(^n^-^1)(|s_(_m_) - s_(_m_-1)|+|s_(m-1) - s_(m-2)|+...[/itex]

    then the term on the left hand side in the inequalities (i.e[itex] s_m in |s_(m) - s_(m-1)|)
    [/itex]
    is the first term in an infinite series and the second one is also the first term in another infinite series.

    therefore the sum of their infinite differences is[itex] 1/(1-r) * |s_2 + s_1|[\itex]

    did I factor out the n correctly? i also still dont know why we have s_2 and s_1


    and they are 1 step apart

    does this sound right? thanks!
     
  6. Nov 15, 2008 #5

    HallsofIvy

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    Well, you need to practice putting { } around subscripts and superscripts but it looks like you have figured out this problem!
     
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