The reciprocal Fibonacci constant is the sum of the reciprocals of the Fibonacci numbers:
\psi=
| infty | |
| \sum | |
| k=1 |
| 1 | |
| Fk |
=
| 1 | |
| 1 |
+
| 1 | |
| 1 |
+
| 1 | |
| 2 |
+
| 1 | |
| 3 |
+
| 1 | |
| 5 |
+
| 1 | |
| 8 |
+
| 1 | |
| 13 |
+
| 1 | |
| 21 |
+ … .
Because the ratio of successive terms tends to the reciprocal of the golden ratio, which is less than 1, the ratio test shows that the sum converges.
The value of is approximately
\psi=3.359885666243177553172011302918927179688905133732...
With terms, the series gives digits of accuracy. Bill Gosper derived an accelerated series which provides digits.[1] is irrational, as was conjectured by Paul Erdős, Ronald Graham, and Leonard Carlitz, and proved in 1989 by Richard André-Jeannin.
Its continued fraction representation is:
\psi=[3;2,1,3,1,1,13,2,3,3,2,1,1,6,3,2,4,362,2,4,8,6,30,50,1,6,3,3,2,7,2,3,1,3,2,...]