SLC Math 1B Topic Reviews (Fall 2023)
Facilitator:
Emily Fleming,
emilyfleming@berkeley.edu
Topic Review 10: The Integral Test
Date:
Wednesday, October 04
Agenda:
•
Introduction
Objectives:
By engaging actively with this session, participants will make progress towards being able to:
i. Employ the integral test to utilize improper integrals in order to prove a series converges or diverges
1. Discern when a series is a good candidate for the integral test
2. Use the Remainder Estimate for the Integral Test to bound the remainder when a sum is estimated by the first
n terms
1) Warm Up:
a)
R
∞
2
1
x
(ln
x
)
3
dx
b) What is the difference between series defined with n as an integer and real valued functions?
2) Determine whether the series is convergent or divergent.
a)
∑
∞
n
=2
1
n
(ln
n
)
3
b)
∑
∞
n
=1
n
5
√
n
6
c)
∑
∞
n
=1
ne
−
n
d)
∑
∞
n
=1
3
n
2
+5
n
−
14
Checkpoint:
What is an example of a series that the integral test cannot be used to determine whether it
converges? Why?
Checkpoint:
Is the sum of the series equal to the sum of the corresponding integral?
3) Use the Remainder Estimate for the Integral Test to estimate the sum of
∑
∞
n
=2
1
n
(ln
n
)
3
for the indicated partial
sum:
a) 5
th
partial sum
b)
n
th
partial sum
4) How many terms are required to have
∑
∞
n
=1
1
n
2
accurate to within 0.0003?
5)
Reflection:
What indicates that a series is a good contender for the integral test?
6) For what values of
p
is the series convergent?
a)
∑
∞
n
=1
1
n
p
b)
∑
∞
n
=1
1
n
(ln
n
)
p
7)
Looking Forward:
Thinking about the connections between series and improper integrals, why might the
p-integrals be useful to know?
