The Chemistry Maths Book, Second Edition

7.2 Sequences 195

EXAMPLES 7.2Limits.

(i)

We have so that.

(ii)

Dividing top and bottom byr

2

,

(iii)

(iv) The Fibonacci sequence (Example 7.1(iv)) is divergent, but a convergent

sequence is obtained from the ratios of consecutive terms,u

r+ 1

2 u

r

. The first 10

terms are 1, 2, 1.5, 1.6666, 1.6, 1.6350, 1.6153, 1.6190. 1.6176, 1.6181, and the

limit of the sequence is

.

Thus, dividing the recurrence relationu

r+ 2

1 = 1 u

r+ 1

1 + 1 u

r

byu

r+ 1

,

u

r+ 2

2 u

r+ 1

1 = 111 + 1 u

r

2 u

r+ 1

and taking the limitr 1 → 1 ∞givesφ 1 = 111 + 112 φ. Thenφ

2

1 − 1 φ 1 − 111 = 10 , with positive

solution. This is identical to the quantity known in geometry as

the ‘golden section’ (ratio).

3

0 Exercises 11–17

φ=+()152

lim ( ).

r

uu

rr

→

+

()

== + ≈

∞

1

φ 1 5 2 1 618034

u

r

r

r

=→= →sin sin

1

00 as ∞

u

rr

rr

r

r

=

++

−+

→→

22 1

11 1

2

2

2

as ∞

u

rr

rr

r

=

++

−+

221

1

2

2

ur

r

→→

1

2

as ∞

u

r

r

=−

1

2

1

u

r

r

r

=

− 2

2

lim ( )

r

u

r

→∞

3

The golden section was known to the Greeks as ‘the division of a straight line in extreme and mean ratio’

(Euclid, ‘The elements’, Book II, Proposition 11, Book VI, Proposition 30). Its present name originated in 15th

Century Italy when it was taken up by artists as a ‘divine proportion’ and used in painting and architecture.

Luca Pacioli (1445–1517) wrote De divina proportione (1509), with illustrations thought to be by Leonardo da

Vinci. Pacioli’s Summa de arithmetica, geometrica, proportioni et proportionalita(Venice, 1494) was one of the

first comprehensive mathematics book to be printed; it contained the first published description of double-

entry bookkeeping. In his ‘Lives of the artists’, Giorgio Vasari (1511–1574) accuses Pacioli of plagiarising the

mathematical works of Piero della Francesca (c.1420–1492).