### Converging Product

In the limit you get the sum of an infinite geometric series. What about an infinite product (1+x)(1+x^2)(1+x^4)... ?

The Fibonacci recurrence relation gives a relationship between the first three terms of the sequence and, if it is also a geometric sequence with common ratio $r$, then this relation gives a quadratic equation in $r$. The question asks for both the result and also its converse to be proved.