7.9 Exercises 223
Section 7.5
Examine the following series for convergence by
Comparison test (useln 1 n 1 < 1 n): 46. 47.
D’Alembert ratio test: 48. 49.
Cauchy integral test: 50. 51.
Section 7.6
Find the radius of convergence of each of the following series:
Write down the first 5 terms of the MacLaurin series of the following functions:
58.(1 1 + 1 x)
123
- 60.(1 1 − 1 x)
− 122
62.sin 12 x
2
- 64.e
− 3 x66.A body with rest mass m
0
and speed vhas relativistic energy
and kinetic energyT 1 = 1 E 1 − 1 m
0
c
2
. Express Tas a power series in vand show that the series
reduces to the nonrelativistic kinetic energy in the limitv 2 c 1 → 10.
67.The equation of state of a gas can be expressed in terms of the series
where theB
iare called virial coefficients. Find the first three coefficients for
(i)the van der Waals equation,
(ii)the Dieterici equation,
(i)Expand each of the following functions as a Taylor series about the given point, and
(ii)find the values of x for which the series converges:
- 69.e
x, 2 70.sin 1 x, π 22 71.ln 1 x, 2
1
1
x
,
p V nb nRTe
an RTV()−=
−p
na
V
- Vnb nRT
−=
2
2
()
pV nRT B T
n
V
iii=
=
∑
0
∞
()
Emc
mc
c
==
−/
2 0
2
22
1 v
e
x
x2− 1
ln( ) 12 22
−+xx
x
1
3 +x
1
1
2
+x
nnnnx
=
∑
−
0
2
1
3
∞
()
mmmmx
=
∑
1
∞
nnx
n
=∑
1
2
∞
nnnx
=
∑
1
∞
rrrx
=
∑
−
0
2
1
∞
()
mmmx
=
∑
0
4
∞
nnn
=∑
2
1
∞
ln
rar
=∑
1
1
∞
rar
=∑
1
1
∞
sas
s
=∑
01
∞
()!
rr
r
=∑
1
3
∞
ln
nn
=
∑
2
1
∞
ln