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Definition
• A __________________ of numbers
Sequences
Lesson 9.1
• Listed according to a given ___________________
• Typically written as a1, a2, … an
• Often shortened to { an }
• Example
• 1, 3, 5, 7, 9, …
• A sequence of ______________ numbers
Finding the nth Term
• We often give an expression of the general
term
• That is used to find a specific term
Sequence As A Function
• Define { an } as a ____________________
•
•
•
•
Domain set of nonnegative _______________
Range subset of the real numbers
Values a1, a2, … called _________of the sequence
Nth term an called the general term
• What is the 5th term of the above sequence?
• Some sequences have limits
• Consider
Converging Sequences
Divergent Sequences
• Note Theorem 9.2 on limits of sequences
• Limit of the sum = sum of limits, etc.
• Finding limit of convergent sequence
•
•
•
•
• Some sequences ___________
• Others just grow __________________
Use table of values
Use ________________
Use knowledge of rational functions
Use ___________________________Rule
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Determining Convergence
Determining Convergence
• Consider
• Manipulate algebraically
• Use l'Hôpital's rule to
_______________________________of the
function
conjugate
expressions
• ___________________and take the limit
Bounded, Monotonic Sequences
• Note we are relating limit of a sequence from the limit
of a ________________ function
Assignment
• Note difference between
• Increasing (decreasing) sequence
• __________________ increasing (decreasing)
sequence
• Table pg 500
• Lesson 9.1
• Page 604
• Exercises 1 – 85 EOO
• Note concept of bounded sequence
• Above
• Below
• Both
Bounded implies ________________
Definition of Series
• Consider summing the terms of an infinite
sequence
Series and Convergence
Lesson 9.2
• We often look at a _______________sum of n
terms
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Definition of Series
Examples
• We can also look at a _____________of
partial sums { Sn }
• Convergent
• The series can _________________________
• Divergent
• The sequence of partial sums converges
• If the sequence { Sn } does not converge, the series
diverges and has no sum
Telescoping Series
• Consider the series
Geometric Series
• Definition
• An infinite series
• The ______________of successive terms in the
series is a ________________
• Note how these could be regrouped and the
end result
• Example
• As n gets large, the series = 1
Properties of Infinite Series
• ______________________
• The series of a sum = the sum of the series
• What is r ?
Geometric Series Theorem
• Given geometric series
(with a ≠ 0)
• Series will
• Use the property
• Diverge when | r | _________
• _______________when | r | < 1
• Examples
• Compound
interest
Or
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Applications
• A pendulum is released through
an arc of length 20 cm
from vertical
• Allowed to swing freely
until stop, each swing 90%
as far as preceding swing past vertical
• How far will it travel
until it comes to rest?
Assignment
• Lesson 9.2
• Page 614
• Exercises 1 – 17 EOO
19 – 24 all
33 – 73 EOO
20 cm
Divergence Test
• Be careful not to confuse
The Integral Test;
p-Series
Lesson 9.3
• Sequence of general terms { ak }
• Sequence of partial sums { Sk }
• We need the distinction for the divergence test
• If
• Then
Convergence Criterion
• Given a series
• If { Sk } is _________________________
• Then the series converges
• Otherwise it diverges
• Note
• Often difficult to apply
• Not easy to determine { Sk } is bounded above
Note this only tells us
about ______________.
must _________ It says nothing about
convergence
The Integral Test
• Given ak = f(k)
• k = 1, 2, …
• f is positive, continuous, _____________for x ≥ 1
• Then
either
• both converge … or
• both _________________
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Try It Out
• Given
p-Series
• Definition
• Does it converge or diverge?
• A series of the form
• Where
p is a _____________________
• Consider
p-Series test
• Converges if _____________
• ___________if 0 ≤ p ≤ 1
Try It Out
Assignment
• Given series
• Lesson 9.3
• Page 622
• Exercises 1 – 41 EOO
43 – 48 all
• Use the p-series test to determine if it
converges or diverges
Direct Comparison Test
• Given
Comparison Tests
Lesson 8.4
• If
converges, then
converges
• What if
• What could you conclude about these?
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Try It on These
• Test for convergence, divergence
• Make comparisons with a geometric series or
p-series
Limit Comparison Test
• Given ak > 0 and bk > 0 for all sufficiently
large k … and …
where L is finite and positive
• Then
either both ___________… or both _________
Limit Comparison Test
Example of Limit Comparison
Strategy for evaluating
• Convergent or divergent?
1. Find series
with _______________ and
general term "essentially same"
2. Verify that this limit exists and is positive
• Find a p-series which is similar
• Consider
3. Now you know that
as
_________________
• Now apply the comparison
Assignment
• Lesson 9.4
• Page 630
• Exercises 5 - 33 EOO
Ratio Test & Root Test
Lesson 9.6
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Ratio Test
• For a series of positive terms
• We realize that the sequence { ak } must be
"rapidly" decreasing towards zero
Use It or Lose It
• Use the ratio test for the following series
• Convergent or divergent?
• Consider the limit of the ratio
• Ratio test says:
• If L < 1 then
converges
• If L > 1 or L is infinite, then
diverges
• If L = 1, the test is inconclusive
The Root Test
• Easiest test we've seen was the divergence
test
• Look at
• If so, the series diverges
• However
convergence
does not guarantee
The Root Test
• Possible to look at
• If L < 1, then
converges
• If L > 1 or if L is infinite then
diverges
• If L = 1, the root test is inconclusive
• Try this out with
Assignment
• Lesson 9.6
• Page 645
• Exercises 5 – 10 all
11 – 47 EOO
Power Series
Lesson 9.8
(Yes we’re doing this before 9.7)
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Definition
Definition
• A power series centered at 0 has the form
• A power series centered at c has the form
• Consider this as an extension of a polynomial
in x
• Consider this as an extension of a polynomial
in x
Examples
• Where are these centered?
Example
Convergence of Power Series
For the power series centered at c
exactly one of the following is true
1. The series converges only for x = c
2. There exists a real number R > 0 such that
the series converges absolutely for |x – c| <
R and diverges for |x – c| > R
3. The series converges absolutely for all x
Dealing with Endpoints
• Consider the power series
• Consider
• What happens at x = 0?
• Converges trivially at x = 0
• Use ratio test
• Use generalized
ratio test for x ≠ 0
• Limit = | x | … converges when | x | < 1
• Try this
• Interval of convergence -1 < x < 1
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Dealing with Endpoints
• Now what about when x = ± 1 ?
Try Another
• Consider
• Again use ratio test
• At x = 1, diverges by the divergence test
• At x = -1, also diverges by divergence test
• Should get
which must be < 1
or -1 < x < 5
• Final conclusion, convergence set is (-1, 1)
• Now check the endpoints, -1 and 5
Power Assignment
Taylor and MacLaurin
Series
• Lesson 9.8
• Page 668
• Exercises 1 – 33 EOO
Lesson 9.7
Taylor & Maclaurin Polynomials
• Consider a function f(x) that can be
differentiated n times on some interval I
• Our goal: find a _____________function M(x)
• which approximates f
• at a number c in its domain
Linear Approximations
• The ____________________is a good
approximation of f(x) for x near a
True value f(x)
Approx. value of f(x)
Centered at c or
____________
f'(a) (x – a)
(x – a)
• Initial requirements
• M(c) = ____________
• ____________ = f '(c)
a
f(a)
x
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Linear Approximations
• Taylor polynomial degree 1
• Approximating f(x) for x near 0
Quadratic Approximations
• For a more accurate approximation to
f(x) = cos x for x near 0
• Use a __________________ function
• We determine
• Consider
• How close are
these?
• f(.05)
• f(0.4)
• At x = 0 we must have
• The functions to agree
• The first and ________________________ to agree
Quadratic Approximations
Quadratic Approximations
• Since
• So
• We have
• Now how
close are
these?
•
•
Taylor Polynomial Degree 2
Higher Degree Taylor Polynomial
• In general we find the approximation of
f(x) for x near 0
• For approximating f(x) for x near 0
• Try for a different function
• Note for f(x) = sin x, Taylor Polynomial of
degree 7
• f(x) = sin(x)
• Let x = 0.3
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Assignment
Improved Approximating
• We can choose some other value for x,
say x = c
• Then for f(x) = ex the
nth degree Taylor polynomial at __________
• Lesson 9.7A
• Page 658
• Exercises 1 – 4 all
5 - 29 odd
Remainder of a Taylor Polynomial
• We need a sense of how accurate our
approximation is
Accuracy in Series
Lesson 9.7B
• Actual
Function
Remainder of a Taylor Polynomial
Approximate
Value
Remainder
Error Calculation
• Error associated with the approximation
• We can determine the maximum error with
the formula
Where …
• M is the bound on the n+1st derivative of f(x)
• d is the number of good digits after the
decimal
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Error Calculation
• When series is all odd terms
• Replace (n + 1) with (2n + 3)
• When series is all even terms
• Replace (n + 1) with (2n + 2)
Error Calculation
• Given f(x) we determine M for the interval
[a, b] spanned by c and x
• Shortcuts
• If f(x) = sin(x) or cos(x), then M = 1
• If f(x) = ex then
• If f(x) = e-x then
• Note
signifies the "ceiling" function, the next
integer beyond the largest value in the interval
Error Calculation
• We will be given
• f(x) … from this we can determine M
• c … the center
• Thus, given any two of x, n, and d you can
determine the other two
Try It Out
• Fifth Maclaurin polynomial for sin x
• Determine P3(0.2)
• Use
to determine
the accuracy of the approximation
• Note remaining shortcuts on handout
Try It Out Some More
• Determine the degree of the Taylor
Polynomial Pn(x) expanded about c = 1 that
should be used to approximate ln(1.3) so that
the error is less than 0.0001
• We are given
Assignment
• Lesson 9.7B
• Page 659
• Exercises 45 – 59 odd
• d
• x
• We seek the value of n
• Note the interval is [1, 1.3]
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