7.2 Sequences 193
(iii) Harmonic sequence:
(iv) Fibonacci sequence (‘series’):
21, 1, 2, 3, 5, 8, 13,= u
r+ 21 = 1 u
r+ 11 + 1 u
r, u
01 = 1 u
11 = 11
0 Exercises 1–10
Limits of sequences
The terms of the harmonic sequence ,
decrease in magnitude as rincreases, and approach the value zero as rtends to
infinity. The quantity
is called the limitof the sequence, and in this case the limit is finite and unique.
Similarly, the sequence
has general termu
r1 = 1 (r 1 − 1 1) 2 rand limit. Thus,
u
101 = 1 0.9, u
1001 = 1 0.99, u
10001 = 1 0.999, =
and whenr 1 = 110
n, the termu
rhas n9’s after the decimal point. When the limit uof a
sequence is finite and unique the sequence is said to convergeto the limit u. When the
lim ( )
r
rr
→
−
=
∞
11
1
2
2
3
3
4
4
5
,,, ,...
lim
r
r→
=
∞1
0
1
1
2
1
3
1
4
,,,,...
1
r
u
r
r
r=, =,,,
1
123 ...
11
2
1
3
1
4
, ,, ,...
2This is the solution to the paria coniculorum, or rabbit problem, given in Fibonacci’s Liber abaci: ‘how many
pairs of rabbits can be bred from one pair in one year if each pair breeds one other pair every month, and they
begin to breed in the second month after birth?’ The Fibonacci sequence has been linked with various patterns of
growth and behaviour in nature.