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MA 126-8C, CALCULUS II
Thursday, November 29, 2012
Name (Print last name first):..........................................
Student Signature:...................................................
TEST IV
Closed book - No calculators!
PART I
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
Determine whether the sequence an = (โ1)n^ n + 8 n^2 + 6
converges or diverges. If it converges,
find its limit.
Answer:.....................
Question 2
Find the limit of the sequence given by an = ln
e^1 /n
. (Your answer must be a number!)
Answer:.....................
Question 3
Determine whether the geometric series
โ^ โ
n=
)n is convergent or divergent. If it is
convergent, find its sum.
Answer:..................
Question 4
Determine whether the infinite series
โ^ โ
n=
(โ1)n^
n + 1 n + 2 is convergent or divergent.
Answer:..................
Question 5
Use the integral test to determine whether the infinite series
โ^ โ
n=
n
ln n
is convergent or
divergent.
Answer:..................
Question 6
Determine whether the alternating series
โ^ โ
n=
(โ1)n^
n^2 + 1 n^4 + 6 is divergent, absolutely conver-
gent, or conditionally convergent.
Answer:..................
Problem 2
This problem has two separate questions. Answer each question!
(a) Find the numerical value of c for which (the infinite series)
โ^ โ
n=
(1 โ c)nโ^1
(b) Find the values of x for which the geometric series
โ^ โ
n=
4 n^ (x + 2)n
converges? Write your answer in interval notation!
Problem 3
Find the radius and interval of convergence of the power series
โ^ โ
n=
(โ2)n โ n (x + 3)n.
Be sure to check any endpoints that exist!
Problem 5
Answer all the following questions.
(a) Find the Maclaurin series representation of the function f (x) = sin(x^2 ). (Hint: The Maclaurin series of sin(x) might prove useful here, if need be!)
(b) Use the series in (a) to evaluate the integral โซ sin(x^2 ) dx
as a power series.
(c) Use the series in (b) to write out the Maclaurin series representation of โซ (^1)
0
sin(x^2 ) dx
(Do not compute and add the terms of your series!)
(d) Find the minimum number of terms you need in the series in (c) to approximate โซ 1
0
sin(x^2 ) dx with an error less than 10โ^3 = 0.001? (Show your work!)
SCRATCH PAPER
(Scratch paper will not be graded!)
