MAT215 Fall 2011
Homework 9
Due Thursday, Dec. 1
Unless stated otherwise, you must justify (at best, prove) all your answers.
1. Do exercises 1, 4 and 5 in Chapter 8.
2. Do exercises 11, 15 (disregarding the last question about vector-valued functions), 16,
26 and 27 in Chapter 5.
3. Let X be the set of all continuous functions from [0, 1] to R.
(a) Let f, g ∈ X. Show that there exists c ∈ [0, 1] such that
|f (c) − g(c)| = sup{|f (x) − g(x)| : x ∈ [0, 1]} .
(b) Define d : X × X → R by
d(f, g) = sup{|f (x) − g(x)| : x ∈ [0, 1]} .
Show that d is a metric.
(c) Let (fn ) be a sequence in X. Show that d(fn , f ) → 0 if and only if fn converges
to f uniformly.
(d) Let (fn ) be a sequence in X given by

1 1

x ∈ 0, 2n+1
∪ 2n , 1
0
1
3
n+2
.
fn (x) = 2 x − 2
x ∈ 2n+1 , 2n+2

3 1
 n+2
−2 x + 4 x ∈ 2n+2 , 2n
Show that fn (x) → 0 pointwise but that (fn ) does not have a convergent subsequence in X.
4. Consider the following inverse function theorem (you do not have to prove it): Suppose
that f : [a, b] → R is strictly increasing and continuous. Then f −1 , defined by
f −1 (y) = x if f (x) = y
is a continuous function on the range (image) of f . Now define the function f : [0, 1] →
R by
(
sin(1/x) x 6= 0
f (x) =
.
0
x=0
Describe the set of all maximal intervals [a, b] ⊂ [0, 1] for which the above inverse
function theorem holds. Also describe their images f ([a, b]) = {f (x) : x ∈ [a, b]}.
5. Define for x ∈ R
∞
∞
2n+1
X
X
x2n
n x
sin x =
(−1)
and cos x =
(−1)n
.
(2n + 1)!
(2n)!
n=0
n=0
(a) Prove that both series above converge absolutely for all x ∈ R.
(b) Prove that for x, y ∈ R,
sin(x + y) = sin x cos y + sin y cos x
and
cos(x + y) = cos x cos y − sin x sin y .
Hint: For this computation you want to try to use similar techniques to those used
to show that ex+y = ex ey in class. For example, you want to use Theorem 3.50 in
the book about products of absolutely convergent series. In addition, you should
use the binomial formula (without proof):
n X
n k n−k
(x + y) =
x y
.
k
k=0
n
Last, observe that sin x consists only of odd powers of x, whereas cos x consists
of only even powers.
(c) Show that
cos x − 1
sin x
= 1 and lim
=0.
x→0 x
x→0
x
Use these and the above identities to compute the derivatives of sin x and of cos x.
lim
(d) Show that for all x, sin2 x + cos2 x = 1.
Hint: Take a derivative.
6. Define π to be the number
2 inf{x > 0 : cos x = 0} .
(Remark: it is possible to show that there exists x such that cos x = 0.)
(a) Show that cos π2 = 0 and sin π2 = 1.
(b) Defining
tan x =
sin x
for x ∈ (−π/2, π/2) ,
cos x
show that tan π4 = 1.
2
(1)
7. Abel’s limit theorem: Suppose that f : (−1, 1] → R is P
a function such that (a) f
n
is continuous at x = 1 and (b) for all x ∈ (−1, 1), f (x) P
= ∞
n=0 an x for some power
series that converges for all x ∈ (−1, 1). If, in addition,
an converges, prove that
∞
X
an = f (1) .
n=0
Hint: For x ∈ (−1, 1) write fn (x) =
Pn
k=0
ak xk and An =
Pn
k=0
ak . Show that
fn (x) = (1 − x)(A0 + · · · + An−1 xn−1 ) + An xn .
Let n → ∞ to get a different representation for f (x). Next denote A =
write
∞
X
f (x) − A = f (x) − (1 − x)
Axn .
P∞
k=0
ak and
n=0
Use the representation of f (x) above to bound this difference for x near 1.
8. For x ∈ (−π/2, π/2) define tan x as in equation (1).
(a) Show that tan x is strictly increasing and compute its derivative.
(b) Show that tan x has an inverse function, called arctan x, defined on all of R,
and compute its derivative. Find a power series expansion for this derivative on
(−1, 1).
(c) Find the power series expansion of arctan x on (−1, 1). You must prove that the
series is equal to arctan x on this interval.
Hint: YouPmay want to use the following fact (shown in class after the break):
n
if f (x) = ∞
n=0 an x is a power series with radius of convergence R then for all
|x| < R,
∞
X
nan xn−1 .
f 0 (x) =
n=1
(d) Use Abel’s limit theorem to show that
π
1 1 1
= 1 − + − + ··· .
4
3 5 7
9. For x ∈ (−1, ∞) define f (x) = log(1 + x) (here log is this the natural logarithm).
(a) Compute the k-th derivative of f for all k.
(b) Compute the Taylor series for f centered at x = 0 and show that it converges to
f (x) for all x ∈ (−1, 1). Use Abel’s limit theorem to show that
log 2 = 1 −
3
1 1 1
+ − + ··· .
2 3 4