MAS221 Analysis, Semester 2 Exercises
Sarah Whitehouse
(Exercises labelled * may be more demanding.)
Chapter 0 Problems: Revision
Question 1
(a) State the definition of convergence of a sequence of real numbers, (an ), to
a limit a.
(b) State the definition of the supremum of a bounded set of real numbers.
(c) Prove that any bounded monotonic increasing sequence converges.
Question 2
Define a sequence (an ) iteratively by
a0 = 0,
an+1 =
an + 1
.
2
(a) Write down the first few terms of this sequence. Can you guess the limit?
(b) Prove that the sequence converges to this limit.
Question 3
Let f : [a, b] → R be a function.
(a) State what is meant by saying that f is continuous at a point x0 ∈ (a, b).
(b) State what is meant by saying that f is differentiable at a point x0 ∈ (a, b).
(c) Prove that if f is differentiable at a point x0 then it is continuous at the
point x0 .
(d) Give an example to show that the converse of the statement in part (c) is
not true.
1
Question 4
Define a function f : R → R by
x2
−x2
f (x) =
x ≥ 0,
x ≤ 0.
• Prove that f is differentiable. Write down the derivative.
• Does the second derivative of f at 0 exist? Justify your answer.
Chapter 1 Problems: Series
P∞
1. Prove that if the series n=1 an converges, then an → 0 as n → ∞. (This
is Proposition 1.3 in the notes; the proof was left as an exercise.)
2. Let
ak =
1
k(k + 1)(k + 2)
By expressing ak in partial fractions, find an expression for the sum
n
X
ak
k=1
Prove that the series
P∞
k=1
ak converges and find the limit.
3. Find an explicit expression for the partial sums of the series
∞
X
1
ln 1 +
.
k
k=1
Deduce that it does not converge.
4. Let (aP
n ) be a sequence of non-zero real numbers and assume that the
∞
series n=1 an converges absolutely. Let (bn ) be a sequence of non-zero
real numbers. Suppose that
an
= 1.
n→∞ bn
lim
Prove that the series
P∞
n=1 bn
converges absolutely.
5. (a) By writing the terms in partial fractions, prove that the series
∞
X
1
n(n + 1)
n=1
converges.
2
(b) By using the result in the previous question, prove that
converges.
P∞
(c) Prove that the series n=1 n1α converges whenever α ≥ 2.
P∞
1
n=1 n2
6. Using any of the tests for convergence, or other reasoning, which of the
following series converge? Justify your answers.
(a)
∞
X
3n + 4n
3n + 5n
n=1
(b)
∞
X
3n + 5n
4n
n=1
(c)
∞
X
1
cos
n
n=1
(d)
∞
X
n=1
√
n2 + 1 −
n
√
n2 − 1
(e)
∞
X
(n!)2
(2n)!
n=1
7. Show that, if |r| < 1, the series
P∞
n=1
nrn−1 converges to
1
(1−r)2 .
Hint: Show that
1 − rn
− nrn ,
1−r
where sn is the n-th partial sum of the series.
(1 − r)sn =
8. Show that
∞
X
tan−1
n=1
Hint: Let a = tan−1
1
2n−1
1
2n2
=
and b = tan−1
tan(a − b) =
π
.
4
1
2n+1
and use the identity
tan a − tan b
.
1 + tan a tan b
9. For each of the following statements, say whether it is true or false, giving
reasons for your answers.
P∞
(a) If limn→∞ an = 0, then n=1 an converges.
3
(b) If limn→∞ (a1 + a2 + · · · + an ) = 0, then
P∞
n=1
an converges.
(c) If an > 0 for all
P∞n and the partial sums sn = a1 + · · · + an are
bounded, then n=1 an converges.
P∞
P∞
(d) If the series n=1 |an | is convergent, then so is n=1 a2n .
Chapter 2 Problems: Integration
1. Let r, s : R → R be step functions.
(a) Show that the product rs : R → R, defined by the formula (rs)(t) =
r(t)s(t), is a step function.
(b) Let a < b. Prove that
b
Z
!2
r(t)s(t) dt
b
Z
≤
a
! Z
r(t) dt
2
a
!
b
2
s(t) dt .
a
(c) Let f, g : [a, b] → R be step functions. Define
Z
kf k =
! 21
b
2
f (t) dt
.
a
Prove that kf + gk ≤ kf k + kgk.
2. Consider the function f : [0, 1] → R defined by f (x) = x2 .
(a) Let n ∈ N. Let rn : [0, 1] → R,
2
2
n−1
1
χ[1/n,2/n) + · · · +
χ[(n−1)/n,1) .
rn (t) = 0χ[0,1/n) +
n
n
Show that rn (t) ≤ f (t) for all t ∈ [0, 1], and calculate the integral
R1
r (t) dt, of the step function rn . You may use without proof the
0 n
formula
1 2 + 2 2 + 3 2 + · · · + n2 =
1
n(n + 1)(2n + 1).
6
(b) Let n ∈ N. Let sn : [0, 1] → R,
sn (t) =
2
2
1
2
χ(0,1/n] +
χ(1/n,2/n] + · · · + 1χ((n−1)/n,1] .
n
n
Show that sn (t) ≥ f (t) for all t ∈ [0, 1], and calculate the integral
R1
s (t) dt, of the step function sn .
0 n
4
(c) Use the above to calculate the integral
Z 1
x2 dx.
0
3. (a) Compute the integral
1
Z
1
p dx.
|x|
−1
(b) Show that the integral
Z
1
−1
1
dx
x2
does not exist.
4. Which of the following functions are Riemann integrable? Justify your
answers.
(a) The function f : [1, 2] → R defined by
f (x) =
exp(sin x)
.
x3 + 5
(b) The function g : R → R defined by g(x) = exp(−x).
(c) The function h : [1, ∞) → R defined by
h(x) =
sin x
.
x2
(d) The function k : [0, 1] → R defined by

0 0 ≤ x < 21 ,




1
1

≤ x < 43 ,

2 2
3
3
7
k(x) = 4 4 ≤ x < 8 ,

7
7
15


 8 8 ≤ x < 16 ,


.. ..
. .
5. (a) Define step functions r, s : R → R by
r(t) = χ[0,1) + eχ[1,2) + e4 χ[2,3] ,
Evaluate the integrals
Z
3
s(t) = eχ[0,1] + e4 χ(1,2] + e9 χ(2,3] .
Z
r(t) dt,
0
3
s(t) dt.
0
2
(b) Why is the function f : [0, 3] → R defined by f (x) = ex Riemann
integrable?
5
(c) Prove that
1 + e + e4 ≤
3
Z
2
ex dx ≤ e + e4 + e9 .
0
6. Let f : R → R be a continuous function, and let a, b : R → R be differentiable functions. Prove that
Z b(x)
d
f (y) dy = b0 (x)f (b(x)) − a0 (x)f (a(x)).
dx a(x)
7. Define a function L : (0, ∞) → R by
Z
x
L(x) =
1
1
dt.
t
Prove the following directly from the definition of L.
(a) L is differentiable, and L0 (x) = x1 .
(b) L(xy) = L(x) + L(y) for all x, y > 0.
(c) L(x/y) = L(x) − L(y) for all x, y > 0.
8. (*) Let f : (0, ∞) → R be a continuous function satisfying the formula
f (xy) = f (x) + f (y) for all x, y > 0.
(a) Prove that f (xa ) = af (x) for all x > 0 and a ∈ R.
[Hint: Prove this first for N, then Z, then Q, and finally extend to R
by continuity.]
(b) Prove that f (x) = f (e) ln(x) for all x > 0.
(c) For the above function L, prove that L(x) = ln x.
9. (a) Compute
Z
2n
1
1
dx.
x
(b) Is the function f : [1, ∞) → R defined by the formula
f (x) =
1
x
integrable? Justify your answer.
10. Compute
Z
1
√
−1
6
1
dx.
1 − x2
11. Let f : [a, b] → R be Riemann integrable. Prove that there is a number
x ∈ [a, b] such that
Z x
Z b
f (t) dt =
f (t) dt.
a
x
12. (a) Let f : [a, b] → R be Riemann integrable. Suppose there are m, M ∈
R such that m ≤ f (x) ≤ M for all x ∈ [a, b]. Prove that there is a
number µ ∈ [m, M ] such that
Z b
f (x) dx = (b − a)µ.
a
(b) Let f : [a, b] → R be continuous. Prove that there is some ξ ∈ [a, b]
such that
Z b
f (x) dx = (b − a)f (ξ).
a
13. Let f : [a, b] → R be continuous, and let g : [a, b] → [0, ∞) be integrable.
Prove that there is some ξ ∈ [a, b] such that
Z b
Z b
f (x)g(x) dx = f (ξ)
g(x) dx.
a
a
Do we need the assumption g(x) ≥ 0? Justify your answer.
Chapter 3 Problems:
Sequences and series of functions
1. (a) State the definition of a sequence of functions fn : R → R converging
uniformly to a function f : R → R.
(b) Prove that if a sequence (fn ) of continuous functions fn : R → R
converges uniformly to a function f : R → R, then the function f is
continuous.
2. Consider the sequence of functions (fn ), where fn : [0, π] → R is defined
by fn (x) = sinn (x). Show that (fn ) converges pointwise. Is (fn ) uniformly
convergent? Justify your answer.
3. For each of the following sequences of functions (fn ) determine the pointwise limit (if it exists) on the indicated interval, and decide whether (fn )
converges uniformly to this limit.
(a) fn (x) = x1/n , x ∈ [0, 1].
(b)
fn (x) =
0
x ≤ n,
x − n x ≥ n,
7
x ∈ R.
(c) fn (x) = ex /xn , x ∈ (1, ∞).
2
(d) fn (x) = e−nx , x ∈ [−1, 1].
2
(e) fn (x) = e−x n, x ∈ R.
4. For each of the following sequences of functions (gn ) find the pointwise
limits, and determine whether they converge uniformly on [0, 1], and on
[0, ∞).
(a) gn (x) = x/n.
(b) gn (x) = xn /(1 + xn ).
(c) gn (x) = xn /(n + xn ).
5. For each of the following sequences of functions (hn ), where hn : [0, 1] → R,
find the pointwise limit, if it exists, and in that case determine whether
the sequence converges uniformly.
(a) hn (x) = (1 − x/n)2 .
(b) hn (x) = x − xn .
Pn
(c) hn (x) = k=0 xk .
6. Define fn : R → R by
n + cos x
.
2n + sin2 x
fn (x) =
(a) Find the pointwise limit of the sequence of functions (fn ).
(b) Show that the sequence (fn ) converges uniformly.
(c) Calculate
7
Z
lim
fn (x) dx.
n→∞
2
7. (a) Let n ∈ N. Show that we can define a continuous function
fn : [0, 1] → R by

x = 0,
 0
1/n
x
−1
fn (x) =
0 < x < 1,
 1 ln x
x = 1.
n
(Note: you only need check continuity at x = 0 and x = 1.)
(b) Does the sequence (fn ) uniformly converge to a limit? Justify your
answer. If you wish, you may assume without proof that each function fn is monotone increasing.
(c) Prove that
Z
lim
1
fn (x) dx = 0.
n→∞
0
8
8. Compute the limits
1
Z
lim
n→∞
0
4
et
dt,
n
Z
lim
n→∞
2
t2−((sin nt)/n) dt,
1
justifying any procedures you use.
9. (a) Find a sequence of differentiable functions (fn ), where fn : [0, 1] → R,
that converges pointwise to a function f : [0, 1] → R, but such that
the sequence of derivatives (fn0 ) does not converge to f 0 .
(b) Is it possible to have a sequence of differentiable functions (fn ), where
fn : [0, 1] → R, that converges uniformly to a function f : [0, 1] → R,
but such that the sequence of derivatives (fn0 ) does not converge to
f 0 ? Justify your answer.
10. Which of the following functions are uniformly continuous? Justify your
answers. Note: I have changed the codomains to R; they didn’t make
much sense before - sorry.
(a) The function f : (0, 2] → R defined by f (x) = 1/x.
(b) The function g : [1, 2] → R defined by g(x) = 1/x.
(c) The function h : [1, ∞) → R defined by h(x) = 1/x.
11. By using the Weierstrass M -test or otherwise, for each of the following
series, determine whether it converges uniformly on R and whether it
converges uniformly on [0, 1].
(a)
∞
X
1
2 + x2
n
n=1
(b)
∞
X
(−1)n x2n+1
(2n + 1)!
n=0
(c) (*) Note: I have added the (*), since this is a bit more challenging.
∞
X
sin(nx)
n
n=1
(d)
∞
X
sin(nx)
n=1
12. (a) Show the series
0 < a < 1.
P∞
n=1
xn converges uniformly for x ∈ [0, a] whenever
9
(b) Does the series converge uniformly on [0, 1) ? Explain.
13. Prove that there is a function f : R → R defined by
f (x) =
∞
X
sin nx
n2
n=1
and that this function is continuous.
Chapter 4 Problems: Applications I
1. Find the radius of each of convergence of the following power series.
(a)
∞
X
n2
xn
2n
+
1
n=0
(b)
1 − 2x + 4x2 − 8x3 + 16x4 − · · ·
(c)
1 + x2 + x4 + x6 + x8 + · · ·
(d)
∞
X
(−1)n x2n
(2n)!
n=0
(e)
∞
X
(nx)3n
n=0
P∞
2. Let
that
n=0
n
an x be a power series with radius of convergence R. Is it true
Z
0
∞
xX
an tn dt =
n=0
∞
X
an n+1
x
n
+1
n=0
whenever |x| < R?
Justify your answer using any theorems you need from the course.
3. By integrating the power series
∞
X
xn =
n=0
1
1−x
between −r and r, show that
1+r
r3
r5
log
=2 r+
+
+ ···
1−r
3
5
whenever 0 < r < 1. Justify each step.
10
4. (a) Express the function
1
1 + x2
as a power series. What is the radius of convergence?
(b) Use part (a) and question 2 to find a power series converging to the
function arctan(x).
(c) Use part (b) to find a series converging to
π
4.
5. Write down a series with radius of convergence 0.
6. Use the power series for exp(x) to prove that the number e is irrational.
Chapter 5 Problems: Sequences in Rk
1. (a) Let (xn ) be a sequence in Rk with limit x. Let α ∈ R. Prove that
the sequence (αxn ) converges to αx.
(b) Let (xn ) and (y n ) be sequences in Rk with limits x and y respectively.
Prove that the sequence (xn + y n ) converges to x + y.
2. For each of the following sequences in R2 determine whether they converge.
If so, find the limit.
(a) an = (1/n, n2 ).
(b) bn = (n, 1/n2 ).
(c) cn = (1/n, 1/n2 ).
(d) dn = (1/n, (−1)n /n).
(e) en =
1
2
n2 (cos n, n
n
+ sin n).
(f) f n = (1/n, (−1) ).
3. For each of the following sequences in R2 determine whether they converge.
If so, find the limit.
(a) The sequence (xn , yn ) defined iteratively by x1 = 1, y1 = 2, and
xn+1 = yn ,
yn+1 = xn .
(b)
an =
1
1+
n
n
, ln n .
(c) The sequence (un , vn ) defined iteratively by u1 = 0, v1 = 1 and
un+1 =
1
(un + vn ),
2
11
vn+1 =
1
(un − vn ).
2
(d) The sequence defined iteratively by u1 = 0, v1 = 1 and
1
(un + vn ),
2
un+1 =
vn+1 = un vn .
4. (*) For each of the following sequences in R2 determine whether they
converge.
(a) The sequence
an =
1+
1
n
n n 1
, 1−
.
n
√
(b) The sequence (xn , yn ) defined iteratively by x1 = 2 3, y1 = 3, and
2
xn+1
=
1
1
+ ,
xn
yn
yn+1 =
√
xn+1 yn .
Hint: prove that the sequence (xn ) is monotonic decreasing and
bounded below, and the sequence (yn ) is monotonic increasing and
bounded above.
5. Call a sequence (an ) in R2 bounded if there is a constant C ≥ 0 such
that |an | ≤ C for all n. Prove that every bounded sequence in R2 has a
convergent subsequence.
6. Which of the following sets are open? Justify your answer.
(a) The interval [0, 1) in R.
(b) The set {x ∈ R | x 6= 0} in R.
(c) The square (0, 1) × (0, 1) in R2 .
(d) The line R × {0} in R2 .
7. Which of the following subsets of R3 are open? Justify your answers.
• A = {(x, y, z) ∈ R3 | x2 + y 2 + z 2 > 1}.
• B = {(x, y, z) ∈ R3 | x > 0, y > 0, z > 0}.
• C = {(x, y, z) ∈ R3 | x = y = 0}.
• D = {(x, y, z) ∈ R3 | x2 + y 2 + z 2 = 1}.
8. We call a set A ⊆ Rk closed if for any sequence (an ) in A that converges
in Rk , the limit of the sequence also lies in A. Prove that the following
sets are closed.
(a) The sets Rk and the empty set.
(b) The one-point set {x} for some x ∈ Rk .
12
(c) The square [0, 1] × [0, 1] in R2 .
9. (a) Prove that a subset A ⊆ Rk is closed if and only if the complement
Rk \A is open.
(b) Give an example of a set that is neither open nor closed.
10. Let z ∈ C be a complex number. Write z n = xn + iyn . What can we
say about convergence of the sequence (xn , yn ) when |z| < 1 and when
|z| > 1 ?
11. (a) Let (an ) be a sequence in Rk with limit a. Prove that any subsequence of (an ) also has limit a.
(b) Suppose a sequence (an ) has a subsequence that converges. Does the
sequence (an ) itself have to converge? Prove that it does or give a
counterexample.
Chapter 6 Problems:
Functions of several variables
1. Define f : [0, 2] × [0, 1] → R2 by
f (x, y) = (−y, x2 + y).
Find and sketch the image of f .
2. Define f : R2 → R by f (x, y) = x2 + y 2 .
(a) Find and sketch the pre-images f −1 (1), f −1 (0), and f −1 (−1).
(b) Find and sketch the pre-image f −1 ((0, 1]).
3. (a) Prove, using the characterisation of continuity in terms of limits
(Proposition 6.6), that the function f : R2 → R defined by f (x, y) =
xy is continuous.
(b) Prove, using the characterisation of continuity in terms of limits
(Proposition 6.6), that if two functions f, g : R2 → R are continuous, then the product f g : R2 → R defined by
(f g)(x, y) = f (x, y)g(x, y),
x, y ∈ R
is continuous. (This will give an alternative proof to that given in
Proposition 6.9 which was direct from the and δ definition.)
4. Using any results you need from the lectures, prove that the function
f : R2 → R defined by the formula
f (x, y) = sin(exy )
is continuous.
13
5. Let f : R2 → R be continuous. Let x, y ∈ R2 . Suppose that there is some
c ∈ R such that f (x) < c < f (y). Prove that there is some z ∈ R2 such
that f (z) = c.
[Hint: You will need the intermediate value theorem for continuous real
functions here (Semester 1, Theorem 4.3.1). Think about defining a function g by g(t) = f (tx + (1 − t)y). ]
6. (*) Prove that a continuous function f : [a, b] × [c, d] → R is bounded, and
takes on its maximum and minimum values.
7. Define a function f : R2 → R by
xy
x2 +y 2
f (x, y) =
0
(x, y) 6= (0, 0),
(x, y) = (0, 0).
(a) Prove that the functions g, h : R → R defined by g(x) = f (x, 0) and
h(x) = f (0, x) are continuous.
(b) Is the function f continuous at (0, 0) ? Justify your answer.
Chapter 7 Problems: Applications II
1. (a) Let t > 0. Show that we can define a continuous function ft : [0, 1] →
R by

x=0
 0
xt −1
ft (x) =
0<x<1
 ln x
t
x=1
(b) State what it means to say that a sequence of functions (gn ), where
gn : [0, 1] → R, converges uniformly to a function g : [0, 1] → R. What
can you say about the sequence (f1/n ) where ft is as in part (a)? (See
question 7 of Chapter 3.)
(c) Prove that
Z 1
lim
ft (x) dx = 0.
t→0
0
You may assume that each ft is monotone increasing.
(d) Let h : [0, 1] × R → R be a function. State a conditions on h which
ensure that the equation
Z
Z 1
d 1
∂h
h(x, t) dx =
(x, t) dx
dt 0
0 ∂t
holds.
(e) Prove that
Z
0
1
xt − 1
dx = ln(t + 1)
ln x
for all t > 0.
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