- Take any number $x$ (edit: x should be positive, heh)
- Add 1 to it $x+1$
- Find its reciprocal $1/(x+1)$
- Repeat from 2
So, taking $x = 1$ to start:
- 1
- 2 (the + 1)
- 0.5 (the reciprocal)
- 1.5 (the + 1)
- 0.666... (the reciprocal)
- 1.666... (the + 1)
- 0.6 (the reciprocal)
- 1.6 (the + 1)
- 0.625
- 1.625
- 0.61584...
- 1.61584...
- 0.619047...
- 1.619047...
- 0.617647058823..
etc.
If we look at just the "step 3"'s (the reciprocals), we get:
- 1
- 0.5
- 0.666...
- 0.6
- 0.625
- 0.61584...
- 0.619047...
- 0.617647058823..
This appears to always converge to 0.61803399... no matter where you start from. I looked up this number and it is often called "The golden ratio" - 1, or $\frac{1+\sqrt{5}}{2}-1$.
- Is there any "mathematical" way to represent the above procedure (or the terms of the second series, of "only reciprocals") as a limit or series?
- Why does this converge to what it does for every starting point $x$?
edit: darn, I just realized that the golden ratio is actually 1.618... and not 0.618...; I edited my answer to change what the result is apparently (golden ratio - 1).
However, I think I could easily make it the golden ratio by taking the +1 "steps" of the original series, instead of the reciprocation steps of the original series:
- 2
- 1.5
- 1.666...
- 1.6
- 1.625
- 1.61584...
- 1.619047...
- 1.617647058823..
which does converge to $\frac{1+\sqrt{5}}{2}-1$
Explaining either of these series is adequate as I believe that explaining one also explains the other.