Page 40 - PRE-U STPM MATHEMATICS (T) TERM 1

Mathematics Term 1 STPM Chapter 2 Sequences and Series
n
The expansion of (1 + x) , n  Q
+
n
We have seen that the binomial expansion of (1 + x) , where n ∈ Z , is a finite series with (n + 1) terms, i.e.
n
(1 + x) = 1 + nx + n(n – 1) x + n (n – 1)(n – 2) x + … + x .
2
3
n
2! 3!
However, if n is any rational number, i.e. n ∈ Q, then the expansion is an infinite series, i.e.
(1 + x) = 1 + nx + n(n – 1) x + n(n – 1)(n – 2) x + … + n(n – 1) … (n – r + 1) x + …
3
2
n
r
2! 3! r!
This series is called the binomial series, and is valid if |x| , 1, i.e. –1 , x , 1. It is used to find approximations,
up to a degree of accuracy very close to its actual value. 2
n
n!
Note: The notation 1 2 = (n – r)! r! is not applicable if n is not a positive integer.
r
Example 38

Expand each of the following expressions as an ascending series in x, up to the term in x . State the range
4
of x such that the expansion is valid.
(a) 1 1 + 1 x 2 — 1 3 (b) (1 – 2x) –3
2

Solution: By using the binomial expansion
2
3
(1 + x) = 1 + nx + n(n – 1) x + n(n – 1)(n – 2) x
n
2! 3!
+ n(n – 1)\(n – 2)(n – 3) x + …
4
4!
(a) Substitute x = 1 x and n = 1 , we get
2 3
1 1 – 1 2 1 1 – 1 21 1 – 2 2
1
1

1
—
2
2
2
3
1 2
3
2
1 1 + 1 x = 1 + 1 1 x + 3 3 2! 1 1 x + 3 3 3! 3 1 1 x

2
2
3 2
2
1 1 – 1 21 1 – 2 21 1 – 3 2
1

2
+ 3 3 3 3 1 1 x + …
4
4! 2
1 – 2 2 1 – 2 21 – 5 2
1
1
2
2
= 1 + 1 x + 3 3 1 1 x + 3 3 3 1 1 x
3
2
6 2 4 6 8
1 – 2 21 – 5 21 – 8 2
1
3 3 3 3 1
2
4
+ 1 x + …
24 16
= 1 + 1 x – 1 x + 5 x – 5 x + …
2
3
4
6 36 648 1944
1
The expansion is valid if | x| , 1, i.e. |x| , 2 or –2 , x , 2.

2
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02 STPM Math T T1.indd 127 3/28/18 4:21 PM