SERIES AND LIMITS
4.15 Divide the series into two series,nodd andneven. Forr= 2 both are absolutely
convergent, by comparison with
∑
n−^2 .Forr= 1 neither series is convergent,
by comparison with∑
n−^1. However, the sum of the two is convergent, by the
alternating sign test or by showing that the terms cancel in pairs.
4.17 The first term has value 0.833 and all other terms are positive.
4.19 |A|^2 (1−r)^2 /(1 +r^2 − 2 rcosφ).
4.21 Use the binomial expansion and collect terms up tox^4. Integrate both sides of
the displayed equation. tanx=x+x^3 /3+2x^5 /15 +···.
4.23 For example,P 5 (x)=24x^4 − 72 x^2 +9. sinh−^1 x=x−x^3 /6+3x^5 / 40 −···.
4.25 Seta=D+δandb=D−δand use the expansion for ln(1±δ/D).
4.27 The limit is 0 form>n,1form=n,and∞form<n.
4.29 (a)−^12 ,^12 ,∞;(b)−4; (c)−1+2/π.
4.31 (a) First approximation 0.886 452; second approximation 0.886 287. (b) Sety=
sinxand re-expressf(x) = 2 as a polynomial equation.y=sin(0.886 287) =
0 .774 730.
4.33 IfS(x)=
∑∞
n=0e−nxevaluateS(x) and considerdS(x)/dx.
E=hν[exp(hν/kT)−1]−^1.4.35 The series expansion is
px
E(
1
3
−
x^2
45