62 Higher Engineering Mathematics
5. Expand ( p + 2q) 11 as far as the fifth term.
⎡
⎣
p 11 + 22 p 10 q + 220 p 9 q 2
+ 1320 p 8 q 3 + 5280 p 7 q 4
⎤
⎦
6. Determine the sixth term of
3 p +
q
3
13
.
[34749 p
8 q
5 ]
7. Determine the middle term of (2a − 5b) 8 .
[700000 a 4 b 4 ]
8. Use the binomial theorem to determine, correct to 4 decimal places:
(a) (1.003) 8 (b) (1.042) 7
[(a) 1.0243 (b) 1.3337]
9. Use the binomial theorem to determine, correct to 5 significant figures:
(a) (0.98)
7 (b) (2.01)
9
[(a) 0.86813 (b) 535.51]
10. Evaluate (4.044) 6 correct to 3 decimal places.
[4373.880]
7.4 Further worked problems on the
binomial series
Problem 11.
(a) Expand
1
(1 +2x) 3 in ascending powers of x as
far as the term in x 3 , using the binomial series.
(b) State the limits of x for which the expansion
is valid.
(a) Using the binomial expansion of (1 + x) n , where
n =−3 and x is replaced by 2x gives:
1
(1 + 2x) 3 = (1 + 2x) −3
= 1 + (−3)(2x) +
(−3)(−4)
2!
(2x) 2
+
(−3)(−4)(−5)
3!
(2x) 3 + · · ·
= 1 − 6x + 24x 2 − 80x 3 +· · ·
(b) The expansion is valid provided |2x| < 1,
i.e. |x| <
1
2
or −
1
2
2
Problem 12.
(a) Expand
1
(4 − x) 2 in ascending powers of x as
far as the term in x 3 , using the binomial
theorem.
(b) What are the limits of x for which the expansion in (a) is true?
(a)
1
(4 − x) 2 =
1
4
1 −
x
4
2 =
1
4 2
1 −
x
4
2
=
1
16
1 −
x
4
−2
Using the expansion of (1 + x) n
1
(4 − x) 2 =
1
16
1 −
x
4
−2
=
1
16
1 + (−2)
−
x
4
+
(−2)(−3)
2!
−
x
4
2
+
(−2)(−3)(−4)
3!
−
x
4
3 + · · ·
=
1
16
1 +
x
2
+
3x 2
16
+
x 3
16
+· · ·
(b) The expansion in (a) is true provided
x
4
< 1,
i.e. |x| < 4 or −4
√
4 + x in ascending powers of x to four terms. Give
the limits of x for which the expansion is valid.
√
4 + x =
4
1 +
x
4
=
√
4
1 +
x
4
= 2
1 +
x
4
1
2
Using the expansion of (1 + x) n ,
2
1 +
x
4
1
2
= 2
1 +
1
2
x
4
+
(1/2)(−1/2)
2!
x
4
2
5. Expand ( p + 2q) 11 as far as the fifth term.
⎡
⎣
p 11 + 22 p 10 q + 220 p 9 q 2
+ 1320 p 8 q 3 + 5280 p 7 q 4
⎤
⎦
6. Determine the sixth term of
3 p +
q
3
13
.
[34749 p
8 q
5 ]
7. Determine the middle term of (2a − 5b) 8 .
[700000 a 4 b 4 ]
8. Use the binomial theorem to determine, correct to 4 decimal places:
(a) (1.003) 8 (b) (1.042) 7
[(a) 1.0243 (b) 1.3337]
9. Use the binomial theorem to determine, correct to 5 significant figures:
(a) (0.98)
7 (b) (2.01)
9
[(a) 0.86813 (b) 535.51]
10. Evaluate (4.044) 6 correct to 3 decimal places.
[4373.880]
7.4 Further worked problems on the
binomial series
Problem 11.
(a) Expand
1
(1 +2x) 3 in ascending powers of x as
far as the term in x 3 , using the binomial series.
(b) State the limits of x for which the expansion
is valid.
(a) Using the binomial expansion of (1 + x) n , where
n =−3 and x is replaced by 2x gives:
1
(1 + 2x) 3 = (1 + 2x) −3
= 1 + (−3)(2x) +
(−3)(−4)
2!
(2x) 2
+
(−3)(−4)(−5)
3!
(2x) 3 + · · ·
= 1 − 6x + 24x 2 − 80x 3 +· · ·
(b) The expansion is valid provided |2x| < 1,
i.e. |x| <
1
2
or −
1
2
2
Problem 12.
(a) Expand
1
(4 − x) 2 in ascending powers of x as
far as the term in x 3 , using the binomial
theorem.
(b) What are the limits of x for which the expansion in (a) is true?
(a)
1
(4 − x) 2 =
1
4
1 −
x
4
2 =
1
4 2
1 −
x
4
2
=
1
16
1 −
x
4
−2
Using the expansion of (1 + x) n
1
(4 − x) 2 =
1
16
1 −
x
4
−2
=
1
16
1 + (−2)
−
x
4
+
(−2)(−3)
2!
−
x
4
2
+
(−2)(−3)(−4)
3!
−
x
4
3 + · · ·
=
1
16
1 +
x
2
+
3x 2
16
+
x 3
16
+· · ·
(b) The expansion in (a) is true provided
x
4
< 1,
i.e. |x| < 4 or −4
√
4 + x in ascending powers of x to four terms. Give
the limits of x for which the expansion is valid.
√
4 + x =
4
1 +
x
4
=
√
4
1 +
x
4
= 2
1 +
x
4
1
2
Using the expansion of (1 + x) n ,
2
1 +
x
4
1
2
= 2
1 +
1
2
x
4
+
(1/2)(−1/2)
2!
x
4
2