72 Higher Engineering Mathematics
Problem 11. Develop a series for sinhxusing
Maclaurin’s series.
f(x)=sinhxf( 0 )=sinh0=
e^0 −e−^0
2
= 0
f′(x)=coshxf′( 0 )=cosh0=
e^0 +e−^0
2
= 1
f′′(x)=sinhxf′′( 0 )=sinh0= 0
f′′′(x)=coshxf′′′( 0 )=cosh0= 1
fiv(x)=sinhxfiv( 0 )=sinh0= 0
fv(x)=coshxfv( 0 )=cosh 0= 1
Substituting in equation (5) gives:
sinhx=f( 0 )+xf′( 0 )+
x^2
2!
f′′( 0 )+
x^3
3!
f′′′( 0 )
+
x^4
4!
fiv( 0 )+
x^5
5!
fv( 0 )+···
= 0 +(x)( 1 )+
x^2
2!
( 0 )+
x^3
3!
( 1 )+
x^4
4!
( 0 )
+
x^5
5!
( 1 )+···
i.e.sinhx=x+
x^3
3!
+
x^5
5!
+···
(as obtained in Section 5.5, page 49)
Problem 12. Produce a power series for cos^22 x
as far as the term inx^6.
From double angle formulae, cos2A=2cos^2 A−1(see
Chapter 17).
from which, cos^2 A=
1
2
( 1 +cos2A)
and cos^22 x=
1
2
( 1 +cos 4x)
From Problem 1,
cosx= 1 −
x^2
2!
+
x^4
4!
−
x^6
6!
+···
hence cos4x= 1 −
( 4 x)^2
2!
+
( 4 x)^4
4!
−
( 4 x)^6
6!
+···
= 1 − 8 x^2 +
32
3
x^4 −
256
45
x^6 +···
Thus cos^22 x=
1
2
( 1 +cos 4x)
=
1
2
(
1 + 1 − 8 x^2 +
32
3
x^4 −
256
45
x^6 +···
)
i.e.cos^22 x= 1 − 4 x^2 +
16
3
x^4 −
128
45
x^6 +···
Now try the following exercise
Exercise 32 Further problemson
Maclaurin’s series
- Determine the first four terms of the power
series for sin2x⎡using Maclaurin’s series.
⎢
⎣
sin2x= 2 x−
4
3
x^3 +
4
15
x^5
−
8
315
x^7 +···
⎤
⎥
⎦
- Use Maclaurin’s series to produce a power
series for cosh 3xas far as the term inx^6.
[
1 +
9
2
x^2 +
27
8
x^4 +
81
80
x^6
]
- Use Maclaurin’stheorem to determine the first
three terms of the power series for ln[ ( 1 +ex).
ln2+
x
2
+
x^2
8
]
- Determine the power series for cos 4tas far as
the term int^6.
[
1 − 8 t^2 +
32
3
t^4 −
256
45
t^6
]
- Expand e
3
2 xin a power series as far as the term
inx^3.
[
1 +
3
2
x+
9
8
x^2 +
9
16
x^3
]
- Develop, as far as the term inx^4 , the power
series for sec2x.
[
1 + 2 x^2 +
10
3
x^4
]
- Expand e^2 θcos3θas far as the term inθ^2 using
Maclaurin’s series.
[
1 + 2 θ−
5
2
θ^2
]
- Determine thefirst threeterms of the series for
sin^2 xby applying Maclaurin’s theorem.
[
x^2 −
1
3
x^4 +
2
45
x^6 ···
]
- UseMaclaurin’s series to determinetheexpan-
sion of( 3 + 2 t)^4.
[
81 + 216 t+ 216 t^2 + 96 t^3 + 16 t^4
]