AMath/PMath 331
1.
1
2 + n2
x
= supxR
Solutions to Assignment 5
1
1
= 2 . Since
2 + n2
x
n
n=1
1
< , the Weierstrass M-test shows
n2
that this sequence converges uniformly on R. (The convergence of
known result from calculus, and can be established by the
AMath/PMath 331
1. (a) Note that S (x) =
1
1+x2
|SxSy |
|x y |
1
2 (1
+
x
)
x2 +1
Solutions to Assignment 6
lies in the interval
1
2 (1
1), 1 (1 + 1) = (0, 1).
2
Similarly, T (x) =
lies in (0, 1], and T (x) < 1 except for x = 0. So by the Mean
Value Theo
AMath/PMath 331
Assignment 1
Due Friday September 23
1. Let an =
n2 n n for n 1.
(a) Compute L := lim an .
n
(b) Estimate the error |L an |, and nd an integer N so that
1
|L an | < 2 106
for all
n N.
2. Let an = sin(log n) for n 1.
1
2
(a) Show that there
AMath/PMath 331
Assignment 2
Due Monday October 3
1.
(a) For each of the following sets, provide a sketch. State whether it is open,
closed or neither. If it is not closed, identify a limit point of the set which is
not in the set. If it is not open, iden
AMath/PMath 331
Assignment 3
Due Friday October 14
1. Consider a function dened on R2 by
0
f (x, y ) = 0
y
sin 2
x
if y 0
.
if y x2
2
if 0 < y < x
(a) Show that f is continuous on R2 \ cfw_(0, 0).
(b) Show that f is not continuous at the origin.
(c) Show
AMath/PMath 331
Assignment 4
Due Friday October 28
1. Let fn (x) = nxenx on [0, ).
(a) Prove that this sequence converges pointwise.
(b) Is the convergence uniform?
x
on [0, ).
1 + nx2
(a) Prove that this sequence converges pointwise.
2. Let gn (x) =
(b)
AMath/PMath 331
Assignment 5
Due Monday November 7
1. Prove that
n=1
x2
1
converges uniformly on the whole real line.
+ n2
2. Show that if f C[0, 1], then Bn f
f
for all n 0.
3. Look at the proof of the Weierstrass Theorem using the function f (x) = x
=
AMath/PMath 331
Assignment 6
Due Monday November 21
1. Let Sx =
1
2
x+
x2 + 1 and T x = tan1 x on R.
(a) Show that |Sx Sy | < |x y | and |T x T y | < |x y | for x, y R.
(b) Show that S and T are not contraction maps.
(c) Show that T has a unique xed point
AMath/PMath 331
Assignment 7
Due Friday December 2
1.
(a) Let u(r, ) = (3 4r2 + r4 ) + (8r2 8r4 ) sin2 + 8r4 sin4 . Compute u(r, ).
(b) Let u(r, ) = log r. Compute u and u(1, ). Explain why u is not a solution of the
heat equation for the boundary functio
AMath/PMath 331
1. (a)
n2 n n =
Solutions to Assignment 1
n2 n n
n2 n + n
n
1
=
=
n2 n + n
n2 n + n
1+ 1
Thus
1
L = lim
n
1
1+
1
n
.
1
n
1
= .
2
(b) Therefore
|L an | =
1
1
2
1+ 1
1
1
n
1
=
1
n
2(1 +
1
1/n
=
1
n)
2 1+
1
1
n
2
1
=
2n 1 +
1
1
n
2.
1
1
and i
AMath/PMath 331
Solutions to Assignment 2
1. (a) (i) A is the union of a spiral starting at (1, 0) and ending up at (0, 0) together with the line
segment from (0, 0) to (1, 0). Both endpoints are included. This is a closed set. It is not open
1
because (1
AMath/PMath 331
Solutions to Assignment 3
1. (a) In the open region cfw_(x, y ) : y < 0 or y > x2 , the function is constant and thus cony
tinuous. In the open region cfw_(x, y ) : 0 < y < x2 , the function sin 2 is a composition
x
of continuous functions
AMath/PMath 331
Solutions to Assignment 4
1. (a) Clearly lim fn (0) = 0. If x > 0, then
so that 1
n
1
+ N < ex .
fn+1 (x)
n+1
=
. Since ex > 1, there is an N
fn (x)
nex
Thus for n N ,
fn+1 (x)
N +1
:= r < 1.
fn (x)
N ex
Hence 0 < fN +m (x) rm fN (x) for m
Final exam subject material - Fall 2014
To prepare for the final exam,
1) Review and learn all definitions and titled theorems, including the ones in the last few lectures
of the course.
2) Study all assignments and their posted solutions.
3) Review the d