Summary of methods for evaluating limits
To find the limit of a continuous functionf(x) at a point x = a, simply substitute the value a into the function noting that ∞0 = 0 and that ∞0 = ∞. The only difficulty occurs when either of the expressions 00 or ∞∞ results. In this case differentiate top and bottom and try again. Continue differentiating until the top and bottom limits are no longer both zero or both infinity. If the undetermined form 0× ∞ occurs then it can always be rewritten as00 or ∞∞.
4.8 EXERCISES
4.8 The N + 1 complex numbers ωm are given by ωm = exp(2πim/N), for m = 0, 1, 2, . . . , N.
(a) Evaluate the following:
(i)
N m=0
ωm, (ii)
N m=0
ω2m, (iii)
N m=0
ωmxm.
(b) Use these results to evaluate:
(i)
N m=0
cos
2πm N
− cos
4πm N
, (ii)
3 m=0
2msin
2πm 3
.
4.9 Prove that
cosθ + cos(θ + α) +· · · + cos(θ + nα) =sin12(n + 1)α
sin12α cos(θ +12nα).
4.10 Determine whether the following series converge (θ and p are positive real numbers):
(a)
∞ n=1
2 sinnθ n(n + 1), (b)
∞ n=1
2 n2, (c)
∞ n=1
1 2n1/2,
(d)
∞ n=2
(−1)n(n2+ 1)1/2 n ln n , (e)
∞ n=1
np n!.
4.11 Find the real values ofx for which the following series are convergent:
(a)
∞ n=1
xn n + 1, (b)
∞ n=1
(sinx)n, (c)
∞ n=1
nx,
(d)
∞ n=1
enx, (e)
∞ n=2
(lnn)x.
4.12 Determine whether the following series are convergent:
(a)
∞ n=1
n1/2 (n + 1)1/2, (b)
∞ n=1
n2 n!, (c)
∞ n=1
(lnn)n nn/2 , (d)
∞ n=1
nn n!. 4.13 Determine whether the following series are absolutely convergent, convergent or
oscillatory:
(a)
∞ n=1
(−1)n n5/2 , (b)
∞ n=1
(−1)n(2n + 1)
n , (c)
∞ n=0
(−1)n|x|n n! ,
(d)
∞ n=0
(−1)n
n2+ 3n + 2, (e)
∞ n=1
(−1)n2n n1/2 .
4.14 Obtain the positive values ofx for which the following series converges:
∞ n=1
xn/2e−n n .
4.15 Prove that
∞ n=2
ln
nr+ (−1)n nr
is absolutely convergent forr = 2, but only conditionally convergent for r = 1.
4.16 An extension to the proof of the integral test (subsection 4.3.2) shows that, iff(x) is positive, continuous and monotonically decreasing, forx≥ 1, and the series f(1) + f(2) +· · · is convergent, then its sum does not exceed f(1) + L, where L is the integral
∞
1
f(x) dx.
Use this result to show that the sumζ(p) of the Riemann zeta series n−p, with p > 1, is not greater than p/(p− 1).
4.17 Demonstrate that rearranging the order of its terms can make a condition-ally convergent series converge to a different limit by considering the series (−1)n+1n−1= ln 2 = 0.693. Rearrange the series as
S =11+13−12+15+17−14+19+111 −16+131 +· · ·
and group each set of three successive terms. Show that the series can then be written
∞ m=1
8m− 3 2m(4m− 3)(4m − 1), which is convergent (by comparison with
n−2) and contains only positive terms. Evaluate the first of these and hence deduce thatS is not equal to ln 2.
4.18 Illustrate result (iv) of section 4.4, concerning Cauchy products, by considering the double summation
S =
∞ n=1
n r=1
1 r2(n + 1− r)3.
By examining the points in thenr-plane over which the double summation is to be carried out, show thatS can be written as
S =
∞ n=r
∞ r=1
1 r2(n + 1− r)3. Deduce thatS≤ 3.
4.19 A Fabry–P´erot interferometer consists of two parallel heavily silvered glass plates;
light enters normally to the plates, and undergoes repeated reflections between them, with a small transmitted fraction emerging at each reflection. Find the intensity of the emerging wave,|B|2, where
B = A(1− r)
∞ n=0
rneinφ,
withr and φ real.
4.8 EXERCISES
4.20 Identify the series
∞ n=1
(−1)n+1x2n (2n− 1)! ,
and then, by integration and differentiation, deduce the valuesS of the following series:
(a)
∞ n=1
(−1)n+1n2 (2n)! , (b)
∞ n=1
(−1)n+1n (2n + 1)!, (c)
∞ n=1
(−1)n+1nπ2n 4n(2n− 1)!, (d)
∞ n=0
(−1)n(n + 1) (2n)! . 4.21 Starting from the Maclaurin series for cosx, show that
(cosx)−2= 1 +x2+2x4 3 +· · · . Deduce the first three terms in the Maclaurin series for tanx.
4.22 Find the Maclaurin series for:
(a) ln
1 +x 1− x
, (b) (x2+ 4)−1, (c) sin2x.
4.23 Writing thenth derivative of f(x) = sinh−1x as f(n)(x) = Pn(x)
(1 +x2)n−1/2,
wherePn(x) is a polynomial (of order n− 1), show that the Pn(x) satisfy the recurrence relation
Pn+1(x) = (1 + x2)Pn(x)− (2n − 1)xPn(x).
Hence generate the coefficients necessary to express sinh−1x as a Maclaurin series up to terms inx5.
4.24 Find the first three non-zero terms in the Maclaurin series for the following functions:
(a) (x2+ 9)−1/2, (b) ln[(2 +x)3], (c) exp(sinx), (d) ln(cosx), (e) exp[−(x − a)−2], (f) tan−1x.
4.25 By using the logarithmic series, prove that if a and b are positive and nearly equal then
lna
b2(a− b) a + b .
Show that the error in this approximation is about 2(a− b)3/[3(a + b)3].
4.26 Determine whether the following functions f(x) are (i) continuous, and (ii) differentiable atx = 0:
(a) f(x) = exp(−|x|);
(b) f(x) = (1− cos x)/x2forx= 0, f(0) =12; (c) f(x) = x sin(1/x) for x= 0, f(0) = 0;
(d) f(x) = [4− x2], where [y] denotes the integer part of y.
4.27 Find the limit asx→ 0 of [√
1 +xm−√
1− xm]/xn, in whichm and n are positive integers.
4.28 Evaluate the following limits:
(a) lim
x→0
sin 3x
sinhx, (b) lim
x→0
tanx− tanh x sinhx− x , (c) lim
x→0
tanx− x
cosx− 1, (d) lim
x→0
cosecx x3 −sinhx
x5
. 4.29 Find the limits of the following functions:
(a) x3+x2− 5x − 2
2x3− 7x2+ 4x + 4, asx→ 0, x → ∞ and x → 2;
(b) sinx− x cosh x
sinhx− x , asx→ 0;
(c)
π/2
x
y cos y− sin y y2
dy, asx→ 0.
4.30 Use Taylor expansions to three terms to find approximations to (a)4√ 17, and (b)3√
26.
4.31 Using a first-order Taylor expansion aboutx = x0, show that a better approxi-mation thanx0to the solution of the equation
f(x) = sin x + tan x = 2 is given byx = x0+δ, where
δ = 2− f(x0) cosx0+ sec2x0
.
(a) Use this procedure twice to find the solution off(x) = 2 to six significant figures, given that it is close tox = 0.9.
(b) Use the result in (a) to deduce, to the same degree of accuracy, one solution of the quartic equation
y4− 4y3+ 4y2+ 4y− 4 = 0.
4.32 Evaluate
limx→0
1 x3
cosecx−1 x−x
6
.
4.33 In quantum theory, a system of oscillators, each of fundamental frequencyν and interacting at temperatureT , has an average energy ¯E given by
E =¯ ∞
n=0nhνe−nx ∞
n=0e−nx ,
wherex = hν/kT , h and k being the Planck and Boltzmann constants, respec-tively. Prove that both series converge, evaluate their sums, and show that at high temperatures ¯E≈ kT , whilst at low temperatures ¯E ≈ hν exp(−hν/kT ).
4.34 In a very simple model of a crystal, point-like atomic ions are regularly spaced along an infinite one-dimensional row with spacingR. Alternate ions carry equal and opposite charges±e. The potential energy of the ith ion in the electric field due to another ion, thejth, is
qiqj
4π0rij
,
whereqi,qjare the charges on the ions andrijis the distance between them.
Write down a series giving the total contributionViof theith ion to the overall potential energy. Show that the series converges, and, ifViis written as
Vi= αe2 4π0R,
4.9 HINTS AND ANSWERS
find a closed-form expression forα, the Madelung constant for this (unrealistic) lattice.
4.35 One of the factors contributing to the high relative permittivity of water to static electric fields is the permanent electric dipole moment,p, of the water molecule.
In an external fieldE the dipoles tend to line up with the field, but they do not do so completely because of thermal agitation corresponding to the temperature, T , of the water. A classical (non-quantum) calculation using the Boltzmann distribution shows that the average polarisability per molecule,α, is given by
α = p
E(cothx− x−1), wherex = pE/(kT ) and k is the Boltzmann constant.
At ordinary temperatures, even with high field strengths (104V m−1or more), x 1. By making suitable series expansions of the hyperbolic functions involved, show thatα = p2/(3kT ) to an accuracy of about one part in 15x−2.
4.36 In quantum theory, a certain method (the Born approximation) gives the (so-called) amplitudef(θ) for the scattering of a particle of mass m through an angle θ by a uniform potential well of depth V0and radiusb (i.e. the potential energy of the particle is−V0within a sphere of radiusb and zero elsewhere) as
f(θ) =2mV0
2K3(sinKb− Kb cos Kb).
Here is the Planck constant divided by 2π, the energy of the particle is 2k2/(2m) andK is 2k sin(θ/2).
Use l’H ˆopital’s rule to evaluate the amplitude at low energies, i.e. whenk and henceK tend to zero, and so determine the low-energy total cross-section.
[ Note: the differential cross-section is given by|f(θ)|2 and the total cross-section by the integral of this over all solid angles, i.e. 2ππ
0|f(θ)|2sinθ dθ. ]
4.9 Hints and answers