Geometric series

Irrational

Rational

Ratio

r2

cs

QSF

a

b

value

approx.

r

estimates

supercluster

a+b*1.5

rat/irr

1 − 2*(1 − r)

(order)

0

=

0

=

0

0

0

0

0

1

=

1

=

1

1

1

1

1

t

=

t

=

1.61803

1.5

0.927

0.859

0.854

1

+

t

=

t2

=

2.61803

2.5

0.955

0.912

0.91

1

+

2

t

=

t3

=

4.23607

4

0.944

0.891

0.888

2

+

3

t

=

t4

=

6.8541

6.5

0.948

0.899

0.896

3

+

5

t

=

t5

=

11.0902

10.5

0.947

0.897

0.894

5

+

8

t

=

t6

=

17.9443

17

0.947

0.897

0.894

8

+

13

t

=

t7

=

29.0344

27.5

0.947

0.897

0.894

13

+

21

t

=

t8

=

46.9787

44.5

0.947

0.897

0.894

sc(2)

21

+

34

t

=

t9

=

76.0132

72

0.947

0.897

0.894

34

+

55

t

=

t10

=

122.992

116.5

0.947

0.897

0.894

sc(3)

55

+

89

t

=

t11

=

199.005

188.5

0.947

0.897

0.894

89

+

144

t

=

t12

=

321.997

305

0.947

0.897

0.894

144

+

233

t

=

t13

=

521.002

493.5

0.947

0.897

0.894

233

+

377

t

=

t14

=

842.999

798.5

0.947

0.897

0.894

377

+

610

t

=

t15

=

1364

1292

0.947

0.897

0.894

610

+

987

t

=

t16

=

2207

2090.5

0.947

0.897

0.894

sc(6)