What is the limit of the Fibonacci sequence?

1 Answer

The Fibonacci sequence is divergent and it's terms tend to infinity.

This fact can be easily seen if you observe that all terms in the Fibonacci sequence are positive and that each term is the sum of the two previous terms, or:

#forall n : F_n > 0, F_(n+2) = F_(n+1) + F_n#

So, every term in the Fibonacci sequence (for #n>2#) is greater then it's predecessor. Also, the ratio at which the terms grow is increasing, meaning that the series is not limited.

There is, however, and interesting result that tells us that:

#lim_(n to oo) F_(n+1)/F_n = varphi#

where #varphi# is the Golden Ratio .