While looking for interesting problems, I was led to a problem involving infinite pulleys.(adsbygoogle = window.adsbygoogle || []).push({});

The problem and solution can be found http://www.feynmanlectures.info/solutions/infinite_pulleys_sol_1.pdf" [Broken]. Now I follow the solution up to the part where he claims [itex]\frac{1}{4^{i}M_i}[/itex] is always vanishing (tends to 0) as i approaches infinity. This does not seem valid.

From [tex]M_{i-1} = \frac{4M_{i}t^{i-1}}{M_i + t^{i-1}}[/tex] we have

[tex]M_{i-1}=\frac{4t^{i-1}}{1 + \frac{t^{i-1}}{M_i}}[/tex]

Since we know [tex]\frac{t^{i-1}}{M_i} \ge 0[/tex] we then have

[tex]M_{i-1} \le 4t^{i-1}[/tex] or equivalently

[tex]M_i \le 4t^i[/tex]

from this we can get

[tex]\frac{1}{4^{i}M_i}\ge \frac{1}{4}\cdot\frac{1}{4^{i}t^{i}}[/tex]

so that [itex]\frac{1}{4^{i}M_i}[/itex] is vanishing only if [itex]\frac{1}{4^{i}t^{i}}[/itex] is. I'm not sure how to determine the actual convergence of the M_{i}s and I might be missing something but this solution does not seem to be valid because of the above point. Does anyone know of a way to perhaps salvage this solution?

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# Feynman's Infinite Pulley Problem

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