MATH 172
Homework 7
Solutions
1. Find an example in Rn where A B = (A B) .
Solution : Let A = (0, 1] and B = (1, 2). Then A B = (0, 1] (1, 2) = (0, 2). So B and A B are open.
Hence B = B = (1, 2) and (A B) = A B = (0, 2). We easily get that A = (0, 1): (0
MATH 172
Homework 5
Solutions
1. Find the MacLaurin series and its radius of convergence for the following functions :
3x4
7 + 2x3
x3
(b)
(1 + x)2
(a)
(c) ln(1 + x2 )
2
Solution : (a) Replacing x by x3 in the series
7
+
xk =
k=0
and then multiplying by
1
MATH 172
Homework 3
Solutions
1. Prove that the following series converge uniformly on the given domain:
(a)
+
k=0
(b)
+
k=1
x2
x
+ 2k
on [4, 3].
x
3k 4 + x4
over R.
Proof : (a) Pick k 0. For all x [4, 3], we have that |x| 4 and |x2 + 2k | 2k and so
x2
No
MATH 172
Homework 6
Solutions
1. Let cfw_ak k1 and cfw_bk k1 be sequences of real numbers such that the series
that the series
+
+
a2 and
k
k=1
+
b2 converge. Prove
k
k=1
ak bk converges absolutely.
k=1
Proof : Put A =
+
a2 and B =
k
k=1
+
b2 .
k
k=1
Let
MATH 172
Homework 2
Solutions
1. Calculate the following limit (prove your work) :
3
lim
n+
1
nx2 + 3
dx
x3 + nx
Solution : For n 1, put
fn : [1, 3] R : x
nx2 + 3
x3 + nx
Put f (x) = x for all x [1, 3]. We prove that cfw_fn n1 converges uniformly to f on
MATH 172
Homework 4
Solutions
1. Calculate the following series :
+
(a)
k=3
+
(b)
k=2
k(1)k
2k
1
k32k+1
Solution : (a) Deriving
+
xk =
k=0
we nd that
+
kxk1 =
k=1
1
1x
for 1 < x < 1
1
(1 x)2
for 1 < x < 1
Multiplying this by x, we get that
+
+
kxk = x
k=1
MATH 172
Homework 9
Solutions
1. For each of the following functions, prove that the limit as (x, y) (0, 0) does not exist.
x2 y 2
x2 + y 2
x2 + y 4
(b) g(x, y) = 2
x + 2y 4
(a) f (x, y) =
Solution : (a) For k R \ cfw_0, consider the path y = kx. We easil
MATH 172
Homework 8
Solutions
)
1
1
,
: n N is an open covering of (0, 1) that does not have a nite subcovering of (0, 1).
n+2 n
(
)
1
1
Proof : Clearly
,
is an open set for all n N.
n+2 n
1
1
Let x (0, 1). Let m be the largest integer strictly less than
MATH 172
Exam 1
Solutions
1. Let E be a non-empty subset of R and fn : E R a function for all n N. Prove that the sequence cfw_fn n1
converges uniformly on E if and only if
> 0 : N N : m, n N, x E : |fn (x) fm (x)| <
Proof : This is the Uniform Cauchy C
MATH 172
Homework 1
Solutions
1. The following sequences of functions converge to some function on the given domain. Find that function and use
the denitions of pointwise and uniform convergence to prove that the convergence is pointwise but not uniform
o