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
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 x
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
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
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,
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 :
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
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 :
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 pr