SOLUTIONS TO PROBLEM SET 9
PETE L. CLARK
All problems except the last are taken from 2.3 of our text.
1) Let cfw_sn be a sequence and cfw_cn and cfw_dn be subsequences given by cn =
s2n and dn = s2n+1 . Assume that limn cn = limn dn = L. Show that
limn
Sequences and Series:
An Introduction to
Mathematical Analysis
by Malcolm R. Adams
c 2010
Contents
1 Sequences
1.1 The general concept of a sequence
1.2 The sequence of natural numbers
1.3 Sequences as functions . . . . . .
1.4 Convergence . . . . . . . .
SPRING 2011 MATH 3100 COURSE SYLLABUS
PETE L. CLARK
Course: Math 3100: Sequences and Series
Instructor: Prof. Pete L. Clark, Ph.D.
Lectures: MWF 12:20pm - 1:10pm
My Oce: Boyd 502
Oce Hours: MWF 11:40am - 12:20pm, and by appointment
Course text: Sequences
PRACTICE PROBLEMS FOR SECOND MATH 3100 MIDTERM
1) Let cfw_an be a sequence of real numbers.
n=1
a) Say what it means for the innite series
n=1
an to converge.
Solution: For n N, put Sn = a1 + . . . + an . Then the innite series converges
if the sequence
PRACTICE PROBLEMS FOR FIRST MATH 3100 MIDTERM
General Comments: These are practice problems. You should regard each individual problem as being a plausible exam problem. Certainly there are too many
problems here to make one hour-long exam (roughly twice
PRACTICE PROBLEMS FOR THIRD MATH 3100 MIDTERM
On Power Series
1) Find all values of x R at which the following power series converge.
2n
( 2 )
a) n=1 3n 3+5 (x1) .
n +4
3n
Solution: This is actually rather tricky, enough for it to be best to apply the
1
MATH 3100 THIRD MIDTERM EXAM: WITH SOLUTIONS
Directions: Please solve all the problems. Calculators are not permitted. You
have 55 minutes. Good luck!
1) Let f (x) = x log x.
a) Find the Taylor series expansion T (x) for f (x) centered at c = 1.
Solution:
MATH 3100 SECOND MIDTERM EXAM: WITH SOLUTIONS
Directions: Please solve all the problems. Calculators are not permitted. You
have 55 minutes. Good luck!
1) a) State the nth term test for convergence.
Solution: Let n an be a convergent real series. Then lim
PRACTICE PROBLEMS FOR SECOND MATH 3100 MIDTERM
1) Let cfw_an be a sequence of real numbers.
n=1
a) Say what it means for the innite series
n=1
an to converge.
Solution: For n N, put Sn = a1 + . . . + an . Then the innite series converges
if the sequence
PRACTICE PROBLEMS FOR SECOND MATH 3100 MIDTERM
General Comments: These are practice problems. You should regard each individual problem as being a plausible exam problem.
The exam will be closed-book and calculators will not be permitted. I would
urge you
Math 3100
Fall 2005
Practice Final Exam
Part I Answer all questions 1. Prove by induction that, for n N,
n
k(k!) = (n + 1)! - 1.
k=1
2. (a) Carefully state the definition of the convergence of a sequence cfw_an to a real number L. 4n 4 (b) Use the defini