138 MATH Winter 2011
Assignment 7
Topics: Sequences, Series, the Integral Test, Estimates of sums
Due: 11 a.m. Friday, March 4.
Instructions:
Print your name and I.D. number at the top of the rst page of your solutions, and
underline your last name.
Submit your solutions in the same order as that of the questions appearing herein.
On collaboration: First attempt the questions on your own, but if you get help
or collaborate with someone, then acknowledge the names of those who helped
you. Any outright copying of assignments will be reported as an act of academic
plagiarism.
On work presentation: Your solutions must have legible handwriting, and must be
presented in clear, concise and logical steps that fully reveal what you are doing.
Questions such as those to be handed in, as well as those that are recommended,
may appear on exams.
To avoid frustration and disappointment, get started on your assignment early.
Do not forget to work on the recommended problems from Stewarts book that are
listed the in on the Schedule and exercise list sheet found on the ACE webpage.
This will give you much needed practice.
Hand in your solutions to the following problems.
1. Suppose {an } is the sequence dened recursively by
a1 = 0
an+1 =
5 + 2an
Prove that {an } converges and nd its limit.
2. Find the sums of the following series.
n
1 + 2n
(a)
(b)
32n+1
4n1
n=0
n=1
1
for n = 1, 2, 3....
3. Find the value of c if
(1 + c)n = 2
n=2
4. For the following, determine whether the series is convergent or divergent.
(a)
n=1
n
en
arctan n
(b)
(c)
n=1
n=1
5.
(a) Use the integral test to show that the series
n=1
1
(1 + n)2
(d)
n=1
n
converges.
n4 + 1
(b) Calculate the partial sum s10 , and estimate the error in using s10 as an approximation to the sum of the series.
6.
(a) Use the integral test to show that the series
n=2
(b) Find n such that the
nth
1
converges.
n(ln n)2
partial sum of the series, sn , estimates the actual
1
.
sum with error at most
100
2
1
n+2