Solve Series Problems with Expert Help | Homework Statement & Equations Included

[Fresh(2^)]

Homework Statement

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

Homework Equations

Not sure:: d = C/pi?

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 C₁ > C₂. 0 < C_n < C_{n - 1} < 1. that let's 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

 

[Filip Larsen]

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 C₁ is 0 and solve some finite sums. Its actually quite interesting how the radius comes out rather simple in the end.