Penn State Altoona
Math 141
The Comparison Tests for Improper Integrals
Improper integrals of the form
a
f (x) dx
Theorem (Comparison Test). Suppose that f and g are continuous functions on [a, ), and suppose
that g(x) > 0 for all suciently large values o
Solutions to Quiz 8
Determine whether the following series are absolutely convergent, conditionally convergent,
or divergent. Specify the test used to reach your conclusion.
1.
n=1
n2
n
sin2 (n)
Solution: The series is divergent. Use the comparison test:
Solutions to Quiz 1
1.
2x3
dx
1 + x2
Solution: Substitute u = 1 + x2 , so du = 2x dx. Then
u1
1
du =
u du
u
u
2
2
1/2
3/2
+ C.
= u3/2 2u1/2 + C = 1 + x2
2 1 + x2
3
3
2.
2x3
dx =
1 + x2
ln x
dx
x2
Solution: Integration by parts:
v=
u = ln x
du =
1
dx
x
d
Solutions to Quiz 11
1. Consider the parametric curve C :
x = t3 3t
y = 3t2 9
(a) Find the x- and y-intercepts.
Solution: x-intercepts: y = 0 = 3t2 9 t = 3. At both t-values we have
x = 0. Thus (0, 0) is the only x-intercept.
y-intercepts: x = 0 = t(t2 3)
Solutions to Quiz 9
1. Determine whether the series
n=2
1
n ln(n99 )
101
converges.
(A) It converges by the Comparison Test with
n=2
1
.
n
(B) It diverges by the Integral Test.
(C) It diverges by the Comparison Test with
n=2
(D) It diverges by the Compari
Solutions to Quiz 10
1. Find parametric equations for the path of a particle that moves along the ellipse
(x 1)2 + 4(y 2)2 = 9
halfway around counterclockwise, starting at the point (1, 1/2).
Solution: Set u = x1 and v = y2 . Then the equation becomes u2
Solutions to Quiz 3
/2
esin x sin(2x) dx
1.
0
Solution: We use the double-angle formula for the sine function and get
/2
/2
e
sin x
esin x 2 sin x cos x dx.
sin(2x) dx =
0
0
Substitute u = sin x, so du = cos x dx:
1
/2
eu 2u du.
esin x 2 sin x cos x dx =
Penn State Altoona
MATH 141
Solutions to Quiz 4
4
1. Evaluate the integral
0
x
dx
x+9
(A) 4 9 ln(13) + 18 ln(3)
/7
2. Evaluate the integral
x cos(7x) dx
0
(C)
2
49
3. Evaluate the integral
(D)
1
3
ln(x + 2) ln x + C
2
2
4. Evaluate the integral
(B)
x1
dx
Penn State Altoona
MATH 141
Solutions to the Challenge Problems on Sequences and Series
Problem 1 (Railroad Problem)
Railroad workers are building a track. On the rst day, they lay 1 mile of track. On the second day,
1
they lay 1 = 1 mile of track. On the
5th Annual Penn State Altoona
Integration Bee Contest
Work Integrals, Win Prizes!
Preliminary Round (Trials): Tuesday, February 4th
In order to be eligible to compete in the final round for a prize you nee
Penn State Altoona
Math 141
Integral Problems
3
x4 ln x dx
1.
1
2/2
2.
0
x2
dx
1 x2
3.
x sin2 x dx
4.
ln(x2 1) dx
5.
sin 2x dx
6.
1+x
dx
1x
/4
tan5 sec3 d
7.
0
8.
9.
1 + ex dx
1 + ex
dx
1 ex
10.
(x + sin x)2 dx
11.
arctan t
dt
t
12.
ln(x + 1)
dx
x2
Penn State Altoona
Math 141
Series Problems
Determine whether the following series are absolutely convergent, conditionally convergent, or divergent.
1.
n sin(1/n)
n=1
Solution: Divergent.
Use the divergence test for series. We have
lim n sin(1/n) = lim
n
Penn State Altoona
Math 141
Solution to a Sequence Problem
1. Dene the sequence cfw_an by
a1 = 5
an+1 = 2 +
1
,
2 + an
n = 1, 2, 3, . . . .
Show that the sequence cfw_an is convergent, and nd lim an .
n
Solution: We compute a few terms of the sequence:
Penn State Altoona
Math 141
Solutions to the Sequence Problems
1. Find a formula for the n-th term an of the sequence
2
3 4
5
, ,
, , .
5
10 17
26
and nd lim an .
n
Solution: We have
an =
This shows that
(1)n+1 (n + 1)
,
(n + 1)2 + 1
1
(n + 1)(1 +
1
(n+1)
Penn State Altoona
Math 141
Solutions to the Comparison Test Problems for Improper Integrals
Determine the convergence or divergence of the following improper integrals.
1.
dx
2x + x2 + 1 + 5
1
Solution: This integral is divergent.
1
Compare to the functi
Penn State Altoona
MATH 141
Challenge Problems Sequences and Series
Problem 1 (Railroad Problem)
Railroad workers are building a track. On the rst day, they lay 1 mile of track. On the second day,
1
they lay 1 = 1 mile of track. On the third day, they lay
Penn State Altoona
Math 141
Sequence Problems
1. Find a formula for the n-th term an of the sequence
2
3 4
5
, ,
, , .
5
10 17
26
and nd lim an .
n
2. Suppose lim an = 3. Find the limit
n
cos(an + enan )
.
n
a2 + 1
n
lim
3. Find the limit, if it exists, o
Penn State Altoona
Math 141
Series Problems
Determine whether the following series are absolutely convergent, conditionally convergent, or divergent.
1.
n sin(1/n)
n=1
2.
n=1
n3
n2 1
+ 2n2 + 5
3.
ln(n)
(1)n
n
n=1
4.
n=1
n
n+1
n2
5.
n!
2 5 8 . . . (3n + 2
Solutions to Quiz 7
Determine whether the series converges or diverges.
1
n
tan
1.
n=1
Solution: The series diverges. Use the Limit Comparison Test, and compare to the
divergent series
n=1
tan
lim
1
n
1
n
n
1
.
n
We have
1
x
tan
= lim
1
x
x
sec2
= lim
1