Continued fractions and the Fibonacci sequence
The convergents of the continued fraction
1+1+1+1+...111 = 1+1+11+11+11+11+11+11+1...
are
12,23,35,58,813,1321,2134,3455,5589,89144,144233,233337,...
where each fraction is composed of successive terms of the Fibonacci sequence.
The sequence of convergents tend to the (so-called) golden ratio ϕ=21(1+√5)=1.618033...