Exam # 2
Spring 2005
MATH 1220-01
Instructor: Oana Veliche
Time: 50 minutes
NAME:
ID#:
INSTRUCTIONS
(1) Fill in your name and your student ID number.
(2) There are 10 problems, each worth 10 points.
(3) Justify all you answers. Correct answers with no justification will not be given any credit.
(4) No books, notes or calculators may be used.
Problem #
1
2
3
4
5
# Points
1
6
7
8
9
10
Total
2
Problem 1.
(a) Decompose the rational function
the numerical values of the coefficients.
(b) Find
Z
2
3
x2
2
dx.
−1
x(3x2
1
into simple fractions. Do not determine
− x)(x2 + 4)
3
Problem 2. Find lim+ x ln x.
x→0
Problem 3. Show that improper integral
Z
0
3
1
dx is divergent.
(x − 1)3
4
Problem 4. Determine whether the sequences converge or diverge. If they converge, find the limit.
Justify all your answers!
(a) an = (−1)n
n+1
.
n
(b) an = e−n sin n.
5
Problem 5. Is the series
∞
X
k=1
limit.
1
1
−
divergent or convergent? If it is convergent, find its
2k − 1
2k
6
Problem 6.
∞
X
22k
is divergent.
(a) Show that the series
4k+1
k=1
(b) Is the series
∞
X
(−2)k+2
k=1
3k−1
divergent or convergent? If it is convergent, find its limit.
7
Problem 7. Determine whether the series
∞
X
n=2
What test(s) do you apply?
1
converges or diverges.
n(ln n)2
8
Problem 8. Determine whether the series
∞
X
n=1
What test(s) do you apply?
2n+1
converges or diverges.
(n + 1)n
9
Problem 9. Consider the series
∞
X
1
2
3
4
5
6
n
= −
+
−
−
+
−··· .
(−1)n+1
(n + 1)(n + 2)
6 12 20 30 42 56
n=1
(a) Is the series conditionally convergent or absolutely convergent ?
(b) Find the smallest number of terms that we need to add in order to estimate the sum of the
1
series with error < .
8
10
Problem 10. Determine whether the following statements are true or false. You do not have to
justify your answers!
(a) If
∞
X
an converges, then lim an = 0.
n→∞
n=1
∞
X
an+1
(b) If an > 0 and lim
= 1, then
an converges.
n→∞ an
n=1
(c) If lim |an | = 1 then, the sequence {an }∞
n=1 is divergent.
n→∞
(d) If n < an for all n ≥ 1, then the sequence {an }∞
n=1 is divergent.