Page 8 - PRE-U STPM MATHEMATICS (T) TERM 1

Mathematics Term 1 STPM Chapter 2 Sequences and Series

Example 3

Determine if each of the following sequences is convergent or divergent.
(a) 3, 3, 3, …, 3, …
1
1
1 2
(b) 1, – , 1 , – , .... , (–1) n – 1 1 , …

2 3 4 n
2
3
(c) 1, r, r , r , …, r n – 1 , … if
(i) |r| , 1, (ii) |r|  1.
Solution: (a) u = 3

lim u = lim 3 = 3.
n → ∞ n n → ∞ 2
The sequence is convergent.
1 2
(b) u = (–1) n – 1 1
n
n
lim u = lim 5 (–1) n – 1 1
1 26
n → ∞ n n → ∞ n
1
= (–1) n – 1 lim 1 2
n → ∞ n
= (–1) n – 1 · 0
= 0
The sequence is convergent.
(c) u = r n – 1
n
lim u = lim (r n – 1 )
n → ∞ n n → ∞
lim u = 0 if |r| , 1
n → ∞ n
and lim u = ∞ if |r|  1.
n → ∞ n
The sequence is convergent if |r| , 1 and divergent if |r|  1.

Properties of the limits of sequences

lim
lim
v = B, then
u = A and
If n → ∞ n n → ∞ n
(a) lim (u ± v ) = lim u ± lim v = A ± B. (b) lim (u · v ) = lim u · lim v = A · B.
n → ∞ n n n → ∞ n n → ∞ n n → ∞ n n n → ∞ n n → ∞ n
lim
u
u
n
(c) lim 1 2 = n → ∞ n = A , provided B ≠ 0.
n → ∞ v lim B
n v
n → ∞ n
Example 4
1
If u and v are the n terms of two sequences where u = 3 and v = respectively, find the limit of each
th
n
n
n
n
n
sequence when n → ∞. Hence, find lim 1 3n + 1 2 .
n → ∞ n
Solution: lim u = lim 3 = 3.
n → ∞ n n → ∞
1
lim v = lim 1 2 = 0

n → ∞ n n → ∞ n
95

02 STPM Math T T1.indd 95 3/28/18 4:21 PM