Brief Study Guide for Chapter 12 -
A (Sections 12.1-12.7) |
Preview For
Chapter 12 |
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Infinite Sequences and
Series |
The study of
(infinite) series is widely regarded as the most difficult topic in all of
calculus. The idea of "adding up" an infinite number of
terms and the "sum of the series" being a finite number
sometimes and not a finite number other times seems to boggle the
imagination. Sequences
seem to be manageable, since the terms of an (infinite) sequence either
approach a number (the limit) sufficiently or don't approach a number
sufficiently, as n goes to infinity. Our experience with limits of
functions extends well to sequences. The
convergence of series to a sum is defined in terms of the convergence of
sequences to a limit. Clarifying the dual use of the word
"converge" in the previous sentence is key in your success in
dealing with series. To determine convergence of series, you must
master the concepts sequences. |
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Calculus 2
Lesson 12.1 Sequences |
| The following topics are the most
important. Typical exercises from Calculus, 5th edition, by James Stewart, are
assigned at the end of each objective. |
| 1. |
Define sequence, limit of a sequence, convergence
of a sequence. P746#1 |
| 2. |
Find the formula for the general term {an}.
P747#11,13 |
| 3. |
Determine whether a sequence converges or
diverges using: |
| 3a. |
"connect the dots", i.e., limit of
f(x) by Chapter
2 methods; P747#17,23,27,37 |
| 3b. |
"connect the dots", i.e., limit of
f(x) by
l'Hopital's rule; P747#27,31 |
| 3c. |
geometric sequence, {a rn};
P747#19 |
| 3d. |
p-sequence, {1/np};
P747#25 |
| 3e. |
sandwich theorem; P747#42 |
| 3f. |
absolute value to zero theorem;
P747#21 |
| 3g. |
bounded monotonic theorem;
P747#61 |
| 3h. |
graphing calculator for directly or recursively defined
sequences. P747#41-46 |
| 4. |
Determine whether a sequence is monotone
increasing or monotone decreasing by either using inequalities or by connecting the dots
and using the derivative . P747#53,54 |
| 5. |
State and explain the completeness
property. Text |
| Maple Assignment:
See Maple Assignment |
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Calculus 2
Lesson 12.2 Series |
| 1. |
Define "series
converges". Text P750 |
| 2. |
Find the sum of two types of
convergent series. |
| 2a. |
partial fractions series (telescoping
series) P756#8,23,33 |
| 2b. |
geometric series P756#3,11,15,17 |
| 3 |
Apply divergence tests. |
| 3a. |
harmonic series Text
P753 |
| 3b. |
nth term test P756#9,21,27,31 |
| 4. |
Express a repeating decimal
as a ratio of integers. P756#39 |
| 5. |
Given the nth partial sum, sn,
of a series, find the terms, an, of the series and the limit of sn.
Conclude whether the series converges and, if so, what the sum of the series
is. P757#49,50 |
| 6. |
Find the sum or difference of two or more series.
P756#28 |
| Maple Assignment:
See Maple Assignment |
|
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Calculus 2
Lesson 12.3 The Integral Test |
| 1. |
Apply positive series tests. |
| 1a. |
integral test P765#1,11,17,28 |
| 1b. |
p-series test P765#3,8,12 |
| 2. |
Estimate the sum of a positive decreasing
convergent series for a given number of terms, then improve the estimate. P766#30 |
| Maple Assignment:
none |
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Calculus 2
Lesson 12.4 Comparison Tests
|
| 1. |
Apply two more positive series tests. |
| 1a. |
comparison test P770#1,2,3,5,13 |
| 1b. |
limit comparison test
P770#11,21,23,25 |
| 2. |
Estimate the sum of a positive decreasing
convergent series, then estimate the error. P771#33 |
| Maple Assignment: none |
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Calculus 2
Lesson 12.5 Alternating Series |
| 1. |
Apply alternating series test.
P775#1,3,5,9,15 |
| 2. |
Estimate the sum of an alternating series, then estimate the
error. P775#21 |
| 3. |
Approximate the sum of alternating convergent series to an
indicated accuracy. P776#27 |
| Maple Assignment:
none |
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Calculus 2
Lesson 12.6 Absolute Convergence |
| 1. |
Define "absolute convergence" and
"conditional convergence". Text P776&777 |
| 2. |
Determine whether a series is absolutely
convergent, conditionally convergent or divergent using among other things, |
| 2a. |
the previous methods, P781#5,7,17 |
| 2b. |
the ratio test, P781#1,3,9,15,31 |
| 2c. |
the root test. P782#23,24 |
| Maple Assignment:
none |
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Calculus 2 Lesson 12.7 Strategy for Testing Series |
| 1. |
Use the strategy to decide which series convergence test to
conduct and use that test to determine the convergence or divergence of a series.
P#784#1,3,5,7,11,13,17,21,27
You may wish to refer to Dr.VanVelsir's Infinite
Series Analysis Tree to see some of the strategies in flow chart form. |
| Maple Assignment:
none |
|
Maple worksheets to be submitted for a grade for Chapter
12 - A
(Sect 12.1-12.7)
|
See Maple Assignment |
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