Reciprocal Fibonacci constant

The reciprocal Fibonacci constant <a href="/facts/%CE%A8/AUdz70Lq">ψ</a> is the sum of the <a href="/facts/Reciprocal_(mathematics)/yczdYyCw">reciprocals</a> of the <a href="/facts/Fibonacci_number/mqxpAUQD">Fibonacci numbers</a>:

 
 
 
 ψ
 =
 
 ∑
 
 k
 =
 1
 
 
 ∞
 
 
 
 
 1
 
 F
 
 k
 
 
 
 
 =
 
 
 1
 1
 
 
 +
 
 
 1
 1
 
 
 +
 
 
 1
 2
 
 
 +
 
 
 1
 3
 
 
 +
 
 
 1
 5
 
 
 +
 
 
 1
 8
 
 
 +
 
 
 1
 13
 
 
 +
 
 
 1
 21
 
 
 +
 ⋯
 .
 
 
 {\displaystyle \psi =\sum _{k=1}^{\infty }{\frac {1}{F_{k}}}={\frac {1}{1}}+{\frac {1}{1}}+{\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{5}}+{\frac {1}{8}}+{\frac {1}{13}}+{\frac {1}{21}}+\cdots .}

Because the ratio of successive terms tends to the reciprocal of the <a href="/facts/Golden_ratio/anloSRmD">golden ratio</a>, which is less than 1, the <a href="/facts/Ratio_test/SpvhbMQF">ratio test</a> shows that the sum <a href="/facts/Convergent_series/DTok170z">converges</a>.
The value of ψ is approximately

 
 
 
 ψ
 =
 3.359885666243177553172011302918927179688905133732
 …
 
 
 {\displaystyle \psi =3.359885666243177553172011302918927179688905133732\dots }
 
 (sequence A079586 in the <a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>).
With k terms, the series gives O(k) digits of accuracy. <a href="/facts/Bill_Gosper/2WTMSU3a">Bill Gosper</a> derived an <a href="/facts/Series_acceleration/XmWMXVFh">accelerated series</a> which provides O(k 2) digits.
ψ is <a href="/facts/Irrational_number/Psv19TET">irrational</a>, as was <a href="/facts/Conjecture/oPWUb7SF">conjectured</a> by <a href="/facts/Paul_Erd%C5%91s/63LoUTe1">Paul Erdős</a>, <a href="/facts/Ronald_Graham/ukKzczTL">Ronald Graham</a>, and <a href="/facts/Leonard_Carlitz/b21l2nBP">Leonard Carlitz</a>, and <a href="/facts/Mathematical_proof/7ECDrU80">proved</a> in 1989 by Richard André-Jeannin.
Its <a href="/facts/Simple_continued_fraction/Wg1cMPxN">simple continued fraction</a> representation is:

 
 
 
 ψ
 =
 [
 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
 ,
 …
 ]
 
 
 
 
 {\displaystyle \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,\dots ]\!\,}
 
 (sequence A079587 in the <a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>).

Reciprocal Fibonacci constant open-in-new

Reciprocal Fibonacci constant