Triangle Circuit

A circuit is built by connecting 1 Ω1\ \Omega wires in a pattern of triangles, as shown above. Point AA is the common vertex of all triangles, and point BB is a vertex of one of the triangles at the end of the pattern.

Let nn be the number of triangles in the pattern. (In the diagram above, n=7n = 7.)

Determine the resistance between points AA and BB in the limit where the pattern of wires grows to infinitely many triangles, i.e. limnRAB\lim_{n\to\infty} R_{AB}

Give your answer to 3 decimal places.

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