Wednesday, March 9
Announcements
If you submitted your Quiz #3 for a regrade, come pick it up
from me (before or after class)
WeBWorK #8 is due at 9pm
Covers material from “Week 8”—see syllabus on course
web page
Solutions to (all versions of) Quiz #4 are online
Quiz #4 papers will be available in the MLC starting
tomorrow
Grades posted in Connect by Friday
Wednesday, March 9
Clicker Questions
Clicker Question 1
Staring a series at a different index
Let {an } be a sequence. Suppose you know that the series
∞
X
an = S converges to some value S. What can you say about
n=1
the series
∞
X
an ?
n=4
A. it converges only if
a1 > a2 > a3 > a4
B. it always converges,
to S − (a1 + a2 + a3 )
C. it always diverges
D. it converges only if
a1 , a2 , a3 are positive
E. there’s not enough
information to tell
Two related sums
The partial sums of
∞
X
an = a4 + a5 + · · ·
n=4
are all (a1 + a2 + a3 ) less than the
partial sums of
∞
X
an = a1 + a2 + a3 + a4 + a5 + · · · .
n=1
Clicker Question 2
Practicing the Integral Test
Determine the convergence or divergence of these two series:
∞
X
1 1
1
1
1
3
I.
n2 e−n
II. + +
+
+
+ ···
5 8 11 14 17
n=1
A. both I. and II.
diverge
B. I. converges
but II. diverges
C. I. diverges but
II. converges
D. both I. and II.
converge
The relevant integrals (check: both
integrands are decreasing)
Z
I.
∞
2 −x3
x e
1
Z
dx = lim
t
t→∞ 1
3 t
lim − 31 e−x 1
t→∞
3
x2 e−x dx
= 0 + 31 e−1 .
=
P
II. equals ∞
n=1 1/(3n + 2):
Z ∞
Z t
1
1
dx = lim
dx
t→∞ 1 3x + 2
1 3x + 2
t
= lim 13 ln |3x + 2| 1 = ∞.
t→∞
Clicker Question 3
The “p-series”
∞
X
1
For which real numbers p does the series
converge?
np
n=1
A. converges for p < 0,
but diverges for p ≥ 0
B. diverges for every p
C. converges for p < 1,
but diverges for p ≥ 1
D. converges for p > 1,
but diverges for p ≤ 1
E. converges for p > 0,
but diverges for p ≤ 0
Using the Integral Test
When p > 0: compare to
Z ∞
Z t
1
dx = lim
x−p dx
p
t→∞
x
1
1−p1
11−p
t
−
= lim
t→∞ 1 − p
1−p
(
1
0 − 1−p , if 1 − p < 0,
=
∞,
if 1 − p > 0.
(p = 1 is handled separately.)
When p ≤ 0, use the Test for
Divergence.