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Help with Series Problem 
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#1
Apr2010, 05:53 AM

P: 5

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 C_{n} :: the expression to be found is convergent. Here also C_{1 }> C_{2}. 0 < C_{n} < C_{n  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 C_{n} 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 


#2
Apr2410, 12:20 PM

PF Gold
P: 960

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 C_{n} (hint: use Pythagoras) and then build a recursive definition of these knowing the height of C_{1} 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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