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Homework Help: Help with Series Problem

  1. Apr 20, 2010 #1
    1. The problem statement, all variables and given/known data

    Not expected to do for Home work but i found the problem interesting. The problem is exactly as stated in Att.


    2. Relevant equations

    Not sure:: d = C/pi?



    3. The attempt at a solution

    Really I don't see a systematic way of approaching this problem, but these are the ideas ! have: Notice that the limit of the of the partial sums must be = 1 ( for n = 1 to infinity; n n--> inf.) which is the radius of the outer circles. So Cn :: the expression to be found is convergent. Here also C1 > C2. 0 < Cn < Cn - 1 < 1. that lets me consider 1 > 1/(n + 1). which encourages .. 1 - 1/ (n +1) as the partial sum taking the lim n --> to inf. i get 1 then adding i get 1/n(n+1) for diameter of Cn Is this reasoning correct ? If not I'd appreciate a prod in right direction.

    thanks guys
    -Orson
    *for some reason Att, is showing up, I used the upload from computer tool but i don't see the attachment. ohh my bad
     

    Attached Files:

    Last edited: Apr 20, 2010
  2. jcsd
  3. Apr 24, 2010 #2

    Filip Larsen

    User Avatar
    Gold Member

    I'm not sure I understand your idea so I am not able to tell if the will be productive or not. However, I do know that you can solve the problem by looking at the geometric relationship between height (above bottom line) and radius of Cn (hint: use Pythagoras) and then build a recursive definition of these knowing the height of C1 is 0 and solve some finite sums. Its actually quite interesting how the radius comes out rather simple in the end.
     
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