Preamble

Let's note R_sil the equivalent resistance of the semi-infinite ladder:


What can be said about the following extended circuit?
































mmmh?





























Well, from one side, we have:


From the other side, the extended circuit is a semi-infinite ladder as well.


So...


R + ( R // R_sil ) + R = R_sil

A necessary condition on R_sil is thus :

R_sil² - 2 R R_sil - 2 R² = 0

+ and // being inner laws on R⁺, R_sil > 0 ; eventually :

R_sil = 2 R / (√ 3 - 1)