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MA 126-6D, CALCULUS II
April 21, 2008
Name (Print last name first):..........................................
Student Signature:...................................................
TEST IV
No calculators are permitted!
PART I - Basic Skills
Each question is worth 5 points.
Part I consists of 6 questions. Clearly write your answer (only) in the space provided after each question. You need not show your work for this part of the test. No partial credit is awarded for this part of the test!
Question 1
Find the length of the โarcโ y = 3 + 2 x, when 0 โค x โค 1. (Your answer must be a real number!)
Answer:.....................
Question 2
Find the limit of the sequence given by an = cos
n
. (Your answer must be a number!)
Answer:.....................
Question 3
Determine whether the geometric series
โ^ โ
n=
)nโ 1 is convergent or divergent. If it is
convergent, find its sum.
Answer:..................
Question 4
Determine whether the infinite series
โ^ โ
n=
n + 1 2 n โ 1 is convergent or divergent.
Answer:..................
Question 5
Determine whether the infinite p-series
โ^ โ
n=
n^6 is convergent or divergent.
Answer:..................
Question 6
Determine whether the alternating series
โ^ โ
n=
(โ1)n n^3
is divergent, absolutely convergent,
or conditionally convergent.
Answer:..................
Problem 2
(a) Find the limit of the convergent sequence defined by
a 1 = 1, an+1 = 3 โ
an
(b) Find the values of x for which the geometric series
โ^ โ
n=
x + 2 5
)nโ 1
converges? Write your answer in interval notation!
Problem 3
(a) Find the numerical value of c for which
โ^ โ
n=
(3 + c)n^
(Hint: Note that the series starts from n = 1.)
(b) Use the Integral Test to determine whether the series
โ^ โ
n=
n(ln n)^5
is convergent or divergent. (Show your work!)
Problem 5
Consider the series (^) โ โ
n=
(โ5)n โ n
Answer all the following questions.
(a) Determine whether the ratio test is conclusive or inconclusive. (Justify your answer!)
(b) Determine whether the series is absolutely convergent, conditionally convergent, or divergent. (Only one of these three choices will be accepted as an answer, and you must justify your choice!)
SCRATCH PAPER
