Exercises

SERIES AND LIMITS
Summary of methods for evaluating limits
To find the limit of a continuous function f(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 as 00 or ∞
∞.
4.8 Exercises
4.1
Sum the even numbers between 1000 and 2000 inclusive.
4.2
If you invest £1000 on the first day of each year, and interest is paid at 5% on
your balance at the end of each year, how much money do you have after 25
years?
4.3
How does the convergence of the series
∞
(n − r)!
n!
n=r
depend on the integer r?
4.4
Show that for testing the convergence of the series
x + y + x2 + y 2 + x3 + y 3 + · · · ,
where 0 < x < y < 1, the D’Alembert ratio test fails but the Cauchy root test is
successful.
4.5
Find the sum SN of the first N terms of the following series, and hence determine
whether the series are convergent, divergent or oscillatory:
(a)
∞
n=1
4.6
ln
n+1
n
,
(b)
∞
(−2)n ,
n=0
∞
(−1)n+1 n
.
3n
n=1
By grouping and rearranging terms of the absolutely convergent series
S=
∞
1
,
n2
n=1
show that
So =
4.7
(c)
∞
3S
1
=
.
2
n
4
n odd
Use the difference method to sum the series
N
n=2
2n − 1
.
2n2 (n − 1)2
144
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
ωm ,
(ii)
m=0
N
ωm2 ,
(iii)
m=0
∞
2 sin nθ
,
n(n
+ 1)
n=1
(d)
∞
n=2
2
(−1) (n + 1)
n ln n
(b)
(c)
∞
n=1
1/2
∞
,
(e)
∞
.
n=1
1
,
2n1/2
np
.
n!
(sin x)n ,
(c)
n=1
∞
enx ,
n=1
∞
nx ,
n=1
(e)
∞
(ln n)x .
n=2
Determine whether the following series are convergent:
(a)
∞
n=1
n1/2
,
(n + 1)1/2
(b)
∞
n2
,
n!
n=1
(c)
∞
(ln n)n
,
nn/2
n=1
(d)
∞
nn
.
n!
n=1
Determine whether the following series are absolutely convergent, convergent or
oscillatory:
(a)
∞
(−1)n
,
n5/2
n=1
(d)
(b)
∞
n=0
4.14
sin 12 (n + 1)α
cos(θ + 12 nα).
sin 12 α
∞
2
,
2
n
n=1
(b)
n
∞
xn
,
n
+1
n=1
(d)
4.13
m=0
2πm
3
Find the real values of x for which the following series are convergent:
(a)
4.12
2m sin
Determine whether the following series converge (θ and p are positive real
numbers):
(a)
4.11
3
(ii)
Prove that
cos θ + cos(θ + α) + · · · + cos(θ + nα) =
4.10
ωm xm .
m=0
(b) Use these results to evaluate:
N 4πm
2πm
− cos
,
cos
(i)
N
N
m=0
4.9
N
n2
∞
(−1)n (2n + 1)
,
n
n=1
(−1)n
,
+ 3n + 2
(e)
(c)
∞
(−1)n |x|n
,
n!
n=0
∞
(−1)n 2n
.
n1/2
n=1
Obtain the positive values of x for which the following series converges:
∞
xn/2 e−n
.
n
n=1
145
SERIES AND LIMITS
4.15
Prove that
∞
ln
n=2
4.16
nr + (−1)n
nr
is absolutely convergent for r = 2, but only conditionally convergent for r = 1.
An extension to the proof of the integral test (subsection 4.3.2) shows that, if f(x)
is positive, continuous and monotonically decreasing, for x ≥ 1, and the series
f(1) + f(2) + · · · is convergent, then its sum does not exceed f(1) + L, where L
is the integral
∞
f(x) dx.
1
4.17
−p
Use this result to show that the sum ζ(p) of the Riemann zeta series
n , with
p > 1, is not greater than p/(p − 1).
Demonstrate that rearranging the order of its terms can make a conditionally convergent series converge to a different limit by considering the series
(−1)n+1 n−1 = ln 2 = 0.693. Rearrange the series as
S=
1
1
+
1
3
−
1
2
+
1
5
+
1
7
−
1
4
+
1
9
+
1
11
−
1
6
+
1
13
+ ···
and group each set of three successive terms. Show that the series can then be
written
∞
m=1
4.18
8m − 3
,
2m(4m − 3)(4m − 1)
−2
which is convergent (by comparison with
n ) and contains only positive
terms. Evaluate the first of these and hence deduce that S is not equal to ln 2.
Illustrate result (iv) of section 4.4, concerning Cauchy products, by considering
the double summation
S=
n
∞ n=1 r=1
1
.
r2 (n + 1 − r)3
By examining the points in the nr-plane over which the double summation is to
be carried out, show that S can be written as
S=
∞ ∞
n=r r=1
4.19
1
.
r2 (n + 1 − r)3
Deduce that S ≤ 3.
A Fabry–Pérot 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
with r and φ real.
146
rn einφ ,
4.8 EXERCISES
4.20
Identify the series
∞
(−1)n+1 x2n
,
(2n − 1)!
n=1
and then, by integration and differentiation, deduce the values S of the following
series:
4.21
(a)
∞
(−1)n+1 n2
,
(2n)!
n=1
(b)
∞
(−1)n+1 n
,
(2n
+ 1)!
n=1
(c)
∞
(−1)n+1 nπ 2n
,
4n (2n − 1)!
n=1
(d)
∞
(−1)n (n + 1)
.
(2n)!
n=0
Starting from the Maclaurin series for cos x, show that
2x4
+ ··· .
3
Deduce the first three terms in the Maclaurin series for tan x.
Find the Maclaurin series for:
1+x
,
(b) (x2 + 4)−1 ,
(c) sin2 x.
(a) ln
1−x
(cos x)−2 = 1 + x2 +
4.22
4.23
Writing the nth derivative of f(x) = sinh−1 x as
f (n) (x) =
Pn (x)
,
(1 + x2 )n−1/2
where Pn (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).
4.24
Hence generate the coefficients necessary to express sinh−1 x as a Maclaurin series
up to terms in x5 .
Find the first three non-zero terms in the Maclaurin series for the following
functions:
(a) (x2 + 9)−1/2 ,
(d) ln(cos x),
4.25
4.26
(b) ln[(2 + x)3 ],
(e) exp[−(x − a)−2 ],
By using the logarithmic series, prove that if a and b are positive and nearly
equal then
a
2(a − b)
ln .
b
a+b
Show that the error in this approximation is about 2(a − b)3 /[3(a + b)3 ].
Determine whether the following functions f(x) are (i) continuous, and (ii)
differentiable at x = 0:
f(x) = exp(−|x|);
f(x) = (1 − cos x)/x2 for x = 0, f(0) = 12 ;
f(x) = x sin(1/x) for x = 0, f(0) = 0;
f(x) = [4 − x2 ], where [y] denotes the integer part of y.
√
√
Find the limit as x → 0 of [ 1 + xm − 1 − xm ]/xn , in which m and n are positive
integers.
Evaluate the following limits:
(a)
(b)
(c)
(d)
4.27
4.28
(c) exp(sin x),
(f) tan−1 x.
147
SERIES AND LIMITS
tan x − tanh x
,
sinh x − x
cosec x sinh x
−
(d) lim
.
3
5
x→0
x
x
sin 3x
,
sinh x
tan x − x
,
(c) lim
x→0 cos x − 1
(a) lim
(b) lim
x→0
4.29
x→0
Find the limits of the following functions:
x3 + x2 − 5x − 2
,
as x → 0, x → ∞ and x → 2;
2x3 − 7x2 + 4x + 4
sin x − x cosh x
(b)
,
as x → 0;
sinh x − x
π/2 y cos y − sin y
dy,
as x → 0.
(c)
y2
x
√
Use √
Taylor expansions to three terms to find approximations to (a) 4 17, and
(b) 3 26.
Using a first-order Taylor expansion about x = x0 , show that a better approximation than x0 to the solution of the equation
(a)
4.30
4.31
f(x) = sin x + tan x = 2
is given by x = x0 + δ, where
δ=
2 − f(x0 )
.
cos x0 + sec2 x0
(a) Use this procedure twice to find the solution of f(x) = 2 to six significant
figures, given that it is close to x = 0.9.
(b) Use the result in (a) to deduce, to the same degree of accuracy, one solution
of the quartic equation
y 4 − 4y 3 + 4y 2 + 4y − 4 = 0.
4.32
Evaluate
lim
x→0
4.33
4.34
1
x3
cosec x −
1
x
−
x
6
.
In quantum theory, a system of oscillators, each of fundamental frequency ν and
interacting at temperature T , has an average energy Ē given by
∞
−nx
n=0 nhνe
Ē = ,
∞
−nx
e
n=0
where x = hν/kT , h and k being the Planck and Boltzmann constants, respectively. Prove that both series converge, evaluate their sums, and show that at high
temperatures Ē ≈ kT , whilst at low temperatures Ē ≈ hν exp(−hν/kT ).
In a very simple model of a crystal, point-like atomic ions are regularly spaced
along an infinite one-dimensional row with spacing R. Alternate ions carry equal
and opposite charges ±e. The potential energy of the ith ion in the electric field
due to another ion, the jth, is
qi qj
,
4π0 rij
where qi , qj are the charges on the ions and rij is the distance between them.
Write down a series giving the total contribution Vi of the ith ion to the overall
potential energy. Show that the series converges, and, if Vi is written as
Vi =
148
αe2
,
4π0 R