There is an infinite ladder-shaped grid of 1Ω resistors connecting adjacent nodes as follows:

What is the electrical resistance between (a) and (b)? Note: in the image, every resistor is 1Ω.

The array displayed can be reduced to an infinite series of equivalent parallel resistors of 3, 5, 7, 9, ........... ohms.

The resistance of the equivalent array will be the reciprocal of the sum of the infinite series 1/3 + 1/5 + 1/7 + 1/9 + .........

Since this series is a subset of the harmonic series, the sum of the infinite series is infinite and the reciprocal of the sum is zero.

The resistance of this array is zero.

Posted by sambat on 2008-02-17 23:47:56