Page 22 - PRE-U STPM MATHEMATICS (T) TERM 1

Mathematics Term 1 STPM Chapter 2 Sequences and Series
lim
S
S = n → ∞ n
∞
= lim 3 a – 1 a 2 4
n
r
n → ∞ 1 – r 1 – r
= a – 1 a 2 lim r n
1 – r 1 – r n → ∞
n
If |r|  1, lim r → ∞, and the series is divergent.
n → ∞
n
If |r| , 1, lim r → 0, and the series is convergent with sum to infinity S = a .
n → ∞ ∞ 1 – r
2
When |r| , 1, the geometric series 1 + a + ar + ar + … is convergent, 2
a
with the sum to infinity S = 1 – r .
∞

Example 18

Find the sum of each of the following series.
(a) 18 – 6 + 2 – … (b) 1 – 1 + 1 – …
4 16

1
1
Solution: (a) 18 – 6 + 2 + … = 18 + 18 – 1 2 + 18 – 1 2 2 + …
3
3
This is a geometric series with 1 term a = 18.
st
Common ratio r = – 1
3
u
|r| = – 1 u = 1 , 1
3
3
a
Sum to infinity S = 1 – r
∞
18
=
1
1 – – 1 2
3
= 54
4
= 13 1
2
1
1
(b) 1 – 1 + 1 – … = 1 + 1 – 1 2 + 1 – 1 2 2 + …
4 16 4 4
This is a geometric series with 1 term a = 1.
st
Common ratio r = – 1
4
1
|r| = |– | = 1 , 1
4 4
a
Sum to infinity S = 1 – r
∞
1
=
1
1 – – 1 2
4
= 4
5
109

02 STPM Math T T1.indd 109 3/28/18 4:21 PM