Page 42 - PRE-U STPM MATHEMATICS (T) TERM 1

Mathematics Term 1 STPM Chapter 2 Sequences and Series

Example 40

Expand 1 + 2x as a series in ascending powers of x, up to the term in x . By substituting x = 1 , find
2
1 – 2x 100
an approximation for 51, stating the number of significant figures your result is accurate to.

— 1
1 + 2x (1 + 2x) 2
Solution: 1 – 2x = — 1
(1 – 2x) 2

— 1 – — 1
2
= (1 + 2x) (1 – 2x) 2 2
1 – 1 2
1
3
2
= 1 + 1 (2x) + 2 1·2 2 (2x) + … 4

2
1 – 1 21 – 3 2
3
1
2
× 1 + – 1 2 (–2x) + 2 1·2 2 (–2x) + … 4
2

1
= 1 + x – 1 x + … 21 1 + x + 3 x + … 2
2
2
2
2
2
= 1 + 2x + 2x + … Ignore x and higher powers of x.
3
When x = 1 , substituting into the expansion
100
1 + 2
100 = 1 + 2 1 1 2 + 2 1 1 2 2 + …
1 – 2 100 100
100
102 = 1 + 1 + 1
98 50 5000
51 = 5101
49 5000
1 51 = 5101
7 5000
51 = 35 707
5000
= 7.1414 (correct to 5 significant figures)

Example 41

1
1
– — 1 3 1 – — —
4
1
Expand (1 + x) 4 in ascending powers of x up to the term in x . Prove that 1 + 2 4 = 5 .
2
2 80
1
– — 1 1 —
4
Using your expansion for (1 + x) 4 and x = 80 , find an approximation for 5 , giving your answer correct
to five decimal places.
129

02 STPM Math T T1.indd 129 3/28/18 4:21 PM