cse547/ams547
PRACTICE Final
Spring 2010
(15 extra points. We will correct ONE problem)
NAME
ID:
Test has similar FORMAT to your real FINAL. It has (and FINAL
will have) two parts.
PART ONE covers problems from homeworks 1-3 AND Lecture notes
(Concrete Ma
cse547/ams547
Midterm 1
Spring 2010
100pts + 10 extra credit
NAME
ID:
ams/cs
There are 5 Problems. Each problem is worth 20pts. There is one extra credit problem (10pts). If needed, use extra pages attached.
PROBLEM 1 Use a summation factor to solve the r
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Question 1
(a) A— B ={113}
(b) B” = {1131517.91111,11_.-}
(c) PM)={mu-1{2}.{31{1.2},{1=3}1{1311121311}
(d) .-1 x A = {(111),(1,2),(113)1(2. 1)(2.12].(2,3)1(3:1)1(3.2)1(3.3)}
Question 2
11
l 11
L113 h. =
E n P g rer +1) 11 +1
31 is true became
1 1 1
Question 1
a) i) 39H) = {6.15591}: {1'}, {9}, {fall}, {€55}, {1'55}: {as 1‘56”
ii) AXE: {(e,d), (6,8), {b,(i), (he), (6,11), {(1,8)}.
1.!) Ne since, for example, {1} and {1,2} E ‘PICCII and are net disjoint.
‘3) L.H.S. = {A n B’) n (C m B’)
= {A n C) n (B
cse547/ams547
MIDTERM 2
Spring 2010
100pts + 10 extra points
NAME
ID:
ams/cs
Each problem 1-5 is worth 20pts.
Useful Formulas sheet is attached!
QUESTION 1
Part 1
Prove that the sequence an = n! grows faster then the sequence bn = cn for any c > 0.
Hint:
cse547/ams547
PRACTICE MIDTERM
(10 extra points)
NAME
ID:
Spring 2010
ams/cs
THE TEST IS WORTH 10 EXTRA PTS. You get 1pt for each of attempted problems. We will correct ONE problem of OUR choice (same for all students) - for up to 5
points
QUESTION 1
Prov
cse547/ams547
PRACTICE MIDTERM 2
Spring 2010
5 extra points
NAME
ID:
ONE PROBLEM WILL BE CORRECTED for 5pts.
QUESTION 1 Prove that
1. the series
n=1
n!
nn
converges
2. but the inharmonic series
(1)n+1
n=1
1
n
conditionally converges.
Hint. Use the follow
Problem 1
Let n and m. be integers.
(a) Prove that if at least one of n or m is even then the product n x m is even.
[b}Prove that if both 11 and H1 are odd then the product n x m is odd.
Answer
(a) Direct proof, based on the general fact that:
An integer