AP Calculus BC Module Three Review Assignment
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Date
Nov 24, 2024
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AP Calculus BC Module Three Review Assignment
1)
Determine the limit of the following sequence:
2)
Determine the sum of the following series. Show your work:
3)
Determine the sum of the following series. Show your work:
4)
State if the series is absolutely convergent, conditionally convergent, or divergent. Justify
your answer.
5)
State whether the following series converge or diverge and justify your answer using one of
the known tests. State the name of the test & show all relevant work. For alternating series,
state whether absolutely or conditionally convergent.
a)
b)
c)
6)
Which of the following series diverge? Choose all that apply. State the name of the tests you
use to make your determination.
a)
b)
c)
d)
e)
7)
Given
a) Determine
.
𝑆
4
b) Using the Alternating Series Estimation Theorem, determine the error in approximating
with
.
c)
What is the exact value for the sum?
d) Is the error bound in b accurate? Explain.
8)
True or False:
T / F a)
If
is the sum of a series of numbers, and the
, then the series
𝑛
∞
lim
→
𝑎
𝑛
= 0
converges.
T / F b)
If a series
converges then,
converges.
T / F c)
If
and
converges, then
converges.
𝑎
𝑛
> 𝑎
𝑛+1
> 0
T / F d)
If a power series
diverges at
, then it also diverges at
.
𝑥 = ?
𝑥 =− ?
9)
What are the interval and radius of convergence for each of the following?
a)
b)
10)
Which of the following cannot be found convergent with the ratio test, but can be found
convergent by comparison with a p-series?
a)
b)
c)
11)
a) Determine the first four non-zero terms and the general term for the Taylor Polynomial
centered about 0.
𝑙𝑛 1 + 𝑥
(
)
b) If the first four terms of the Taylor series are used to determine its infinite sum, what is
the resultant error?
c)
Determine the first four terms for the Taylor Polynomial
centered about 1.
𝑙𝑛 1 + 𝑥
(
)
d) Using your result in a, determine the first four terms and the
general term
for the Taylor
Polynomial
.
𝑙𝑛 1 + 𝑥
2
(
)
e) Using your result in a, determine the first four terms and the
general term
for the Taylor
Polynomial
.
𝑥 𝑙𝑛
1 + 𝑥
2
(
)
f)
Using your result in d, determine the first four terms and the
general term
for the Taylor
Polynomial
.
?
?𝑥
𝑙𝑛 1 + 𝑥
2
(
)
[
]
g) Using your results in d, determine
with an error bound of 0.01. Show all
0
1
∫ 𝑙𝑛 1 + 𝑥
2
(
)
?𝑥
work.
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