Fibonacci Convergence


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The table below shows how the ratios of the successive numbers in the Fibonacci series quickly converge on Phi or , which to the first 15 places is:

1.618033988749895 . . .

After the 40th number in the series, the ratio is accurate to 15 decimal places.


Sequence in the series Resulting Fibonacci number (the sum of the two numbers before it) Ratio of each number to the one before it (this estimates phi or ) Difference from

0

0

1 1
2 1 1.000000000000000 +0.618033988749895
3 2 2.000000000000000 -0.381966011250105
4 3 1.500000000000000 +0.118033988749895
5 5 1.666666666666667 -0.048632677916772
6 8 1.600000000000000 +0.018033988749895
7 13 1.625000000000000 -0.006966011250105
8 21 1.615384615384615 +0.002649373365279
9 34 1.619047619047619 -0.001013630297724
10 55 1.617647058823529 +0.000386929926365
11 89 1.618181818181818 -0.000147829431923
12 144 1.617977528089888 +0.000056460660007
13 233 1.618055555555556 -0.000021566805661
14 377 1.618025751072961 +0.000008237676933
15 610 1.618037135278515 -0.000003146528620
16 987 1.618032786885246 +0.000001201864649
17 1,597 1.618034447821682 -0.000000459071787
18 2,584 1.618033813400125 +0.000000175349770
19 4,181 1.618034055727554 -0.000000066977659
20 6,765 1.618033963166707 +0.000000025583188
21 10,946 1.618033998521803 -0.000000009771909
22 17,711 1.618033985017358 +0.000000003732537
23 28,657 1.618033990175597 -0.000000001425702
24 46,368 1.618033988205325 +0.000000000544570
25 75,025 1.618033988957902 -0.000000000208007
26 121,393 1.618033988670443 +0.000000000079452
27 196,418 1.618033988780243 -0.000000000030348
28 317,811 1.618033988738303 +0.000000000011592
29 514,229 1.618033988754323 -0.000000000004428
30 832,040 1.618033988748204 +0.000000000001691
31 1,346,269 1.618033988750541 -0.000000000000646
32 2,178,309 1.618033988749648 +0.000000000000247
33 3,524,578 1.618033988749989 -0.000000000000094
34 5,702,887 1.618033988749859 +0.000000000000036
35 9,227,465 1.618033988749909 -0.000000000000014
36 14,930,352 1.618033988749890 +0.000000000000005
37 24,157,817 1.618033988749897 -0.000000000000002
38 39,088,169 1.618033988749894 +0.000000000000001
39 63,245,986 1.618033988749895 -0.000000000000000
40 102,334,155 1.618033988749895 +0.000000000000000

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