72 Higher Engineering Mathematics
Problem 11. Develop a series for sinh x using
Maclaurin’s series.
f (x) = sinh x
f(0) = sinh 0 =
e 0 − e −0
2
= 0
f (x) = cosh x
f (0) = cosh 0 =
e
0
+ e
−0
2
= 1
f (x) = sinh x f (0) = sinh 0 = 0
f (x) = cosh x f (0) = cosh 0 = 1
f iv (x) = sinh x f iv (0) = sinh 0 = 0
f v (x) = cosh x f v (0) = cosh 0 = 1
Substituting in equation (5) gives:
sinh x = f (0) + x f (0) +
x 2
2!
f (0) +
x 3
3!
f (0)
+
x 4
4!
f iv (0) +
x 5
5!
f v (0) + · · ·
= 0 + (x)(1) +
x 2
2!
(0) +
x 3
3!
(1) +
x 4
4!
(0)
+
x 5
5!
(1) + · · ·
i.e. sinh x = x +
x 3
3!
+
x 5
5!
+ · · ·
(as obtained in Section 5.5, page 49)
Problem 12. Produce a power series for cos 2 2x
as far as the term in x
6 .
From double angle formulae, cos 2 A = 2 cos 2 A − 1 (see
Chapter 17).
from which,
cos 2 A =
1
2
(1 + cos 2 A)
and
cos
2 2x =
1
2
(1 + cos 4x)
From Problem 1,
cos x = 1 −
x 2
2!
+
x 4
4!
−
x 6
6!
+ · · ·
hence
cos 4x = 1 −
(4x) 2
2!
+
(4x) 4
4!
−
(4x) 6
6!
+ · · ·
= 1 − 8x 2 +
32
3
x 4 −
256
45
x 6 + · · ·
Thus cos
2 2x =
1
2
(1 + cos 4x)
=
1
2
1 + 1 − 8x
2
+
32
3
x
4
−
256
45
x
6
+ · · ·
i.e. cos
2 2x = 1− 4x
2
+
16
3
x
4
−
128
45
x
6
+· · ·
Now try the following exercise
Exercise 32 Further problems on
Maclaurin’s series
1. Determine the first four terms of the power
series for sin 2x using Maclaurin’s series.
⎡
⎢
⎣
sin 2x = 2x −
4
3
x 3 +
4
15
x 5
−
8
315
x 7 + · · ·
⎤
⎥
⎦
2. Use Maclaurin’s series to produce a power
series for cosh 3x as far as the term in x 6 .
1 +
9
2
x 2 +
27
8
x 4 +
81
80
x 6
3. Use Maclaurin’s theorem to determine the first
three terms of the power series for ln(1 + e x ).
ln 2 +
x
2
+
x 2
8
4. Determine the power series for cos 4t as far as
the term in t 6 .
1 − 8t 2 +
32
3
t 4 −
256
45
t 6
5. Expand e
3
2 x in a power series as far as the term
in x 3 .
1 +
3
2
x +
9
8
x 2 +
9
16
x 3
6. Develop, as far as the term in x 4 , the power
series for sec 2x.
1 + 2x 2 +
10
3
x 4
7. Expand e 2θ cos 3θ as far as the term in θ 2 using
Maclaurin’s series.
1 + 2θ −
5
2
θ
2
8. Determine the first three terms of the series for
sin
2 x by applying Maclaurin’s theorem.
x 2 −
1
3
x 4 +
2
45
x 6 · · ·
9. Use Maclaurin’s series to determine the expansion of (3 + 2t ) 4 .
81 + 216t + 216t 2 + 96t 3 + 16t 4
Problem 11. Develop a series for sinh x using
Maclaurin’s series.
f (x) = sinh x
f(0) = sinh 0 =
e 0 − e −0
2
= 0
f (x) = cosh x
f (0) = cosh 0 =
e
0
+ e
−0
2
= 1
f (x) = sinh x f (0) = sinh 0 = 0
f (x) = cosh x f (0) = cosh 0 = 1
f iv (x) = sinh x f iv (0) = sinh 0 = 0
f v (x) = cosh x f v (0) = cosh 0 = 1
Substituting in equation (5) gives:
sinh x = f (0) + x f (0) +
x 2
2!
f (0) +
x 3
3!
f (0)
+
x 4
4!
f iv (0) +
x 5
5!
f v (0) + · · ·
= 0 + (x)(1) +
x 2
2!
(0) +
x 3
3!
(1) +
x 4
4!
(0)
+
x 5
5!
(1) + · · ·
i.e. sinh x = x +
x 3
3!
+
x 5
5!
+ · · ·
(as obtained in Section 5.5, page 49)
Problem 12. Produce a power series for cos 2 2x
as far as the term in x
6 .
From double angle formulae, cos 2 A = 2 cos 2 A − 1 (see
Chapter 17).
from which,
cos 2 A =
1
2
(1 + cos 2 A)
and
cos
2 2x =
1
2
(1 + cos 4x)
From Problem 1,
cos x = 1 −
x 2
2!
+
x 4
4!
−
x 6
6!
+ · · ·
hence
cos 4x = 1 −
(4x) 2
2!
+
(4x) 4
4!
−
(4x) 6
6!
+ · · ·
= 1 − 8x 2 +
32
3
x 4 −
256
45
x 6 + · · ·
Thus cos
2 2x =
1
2
(1 + cos 4x)
=
1
2
1 + 1 − 8x
2
+
32
3
x
4
−
256
45
x
6
+ · · ·
i.e. cos
2 2x = 1− 4x
2
+
16
3
x
4
−
128
45
x
6
+· · ·
Now try the following exercise
Exercise 32 Further problems on
Maclaurin’s series
1. Determine the first four terms of the power
series for sin 2x using Maclaurin’s series.
⎡
⎢
⎣
sin 2x = 2x −
4
3
x 3 +
4
15
x 5
−
8
315
x 7 + · · ·
⎤
⎥
⎦
2. Use Maclaurin’s series to produce a power
series for cosh 3x as far as the term in x 6 .
1 +
9
2
x 2 +
27
8
x 4 +
81
80
x 6
3. Use Maclaurin’s theorem to determine the first
three terms of the power series for ln(1 + e x ).
ln 2 +
x
2
+
x 2
8
4. Determine the power series for cos 4t as far as
the term in t 6 .
1 − 8t 2 +
32
3
t 4 −
256
45
t 6
5. Expand e
3
2 x in a power series as far as the term
in x 3 .
1 +
3
2
x +
9
8
x 2 +
9
16
x 3
6. Develop, as far as the term in x 4 , the power
series for sec 2x.
1 + 2x 2 +
10
3
x 4
7. Expand e 2θ cos 3θ as far as the term in θ 2 using
Maclaurin’s series.
1 + 2θ −
5
2
θ
2
8. Determine the first three terms of the series for
sin
2 x by applying Maclaurin’s theorem.
x 2 −
1
3
x 4 +
2
45
x 6 · · ·
9. Use Maclaurin’s series to determine the expansion of (3 + 2t ) 4 .
81 + 216t + 216t 2 + 96t 3 + 16t 4