224 Chapter 7Sequences and series
Section 7.7
- (i) Find the MacLaurin expansion of the function(8 1 + 1 x)
123
up to terms inx
4
. (ii)Use
this expansion to find an approximate value of. (iii)Use this value and Taylor’s
theorem for the remainder to compute upper and lower bounds to the value of.
Find the limits:
77.The energy density of black-body radiation at temperature Tis given by the Planck
formula
where λis the wavelength. Show that the formula reduces to the classical Rayleigh–Jeans
lawρ 1 = 18 πkT 2 λ
4
(i) for long wavelengths(λ 1 → 1 ∞),(ii) if Planck’s constant is set to zero
(h 1 → 1 0).
Section 7.8
78.Find the Cauchy product of the power series expansions of sin 1 xand cos 1 x, and show
that it is equal to.
79.Differentiate the power series expansion of sin 1 xand show that the result is cos 1 x.
80.Integrate the power series expansion of sin 1 xand show that the result isC 1 − 1 cos 1 x, where
Cis a constant.
1
2
sin 2 x
ρλ
λ
λ()=−[ ]
−
8
1
5
1
πhc
e
hc kTlim
ln
x
x
x
→
−
1
2
1
lim
cos
x
ee
x
xx→
+−
−
−
0
2
1
lim
tan sin
x
xx
x
→
−
0
3
lim
x
e
x
x→
−
0
1
9
3
9