Sequences Lesson 8.1

Definition A __________________ of numbers Listed according to a given ___________________ Typically written as a 1, a 2, … a n Often shortened to { a n } Example 1, 3, 5, 7, 9, … A sequence of ______________ numbers

Finding the n th Term We often give an expression of the general term That is used to find a specific term What is the 5 th term of the above sequence?

Sequence As A Function Define { a n } as a ____________________ Domain set of nonnegative _______________ Range subset of the real numbers Values a 1, a 2, … called _________of the sequence N th term a n called the general term Some sequences have limits Consider

Converging Sequences Note Theorem 9.2 on limits of sequences Limit of the sum = sum of limits, etc. Finding limit of convergent sequence Use table of values Use ________________ Use knowledge of rational functions Use ___________________________Rule

Divergent Sequences Some sequences ___________ Others just grow __________________

Determining Convergence Manipulate algebraically ___________________and take the limit conjugate expressions

Determining Convergence Consider Use l'Hôpital's rule to _______________________________of the function Note we are relating limit of a sequence from the limit of a ________________ function

Bounded, Monotonic Sequences Note difference between Increasing (decreasing) sequence __________________ increasing (decreasing) sequence Table pg 500 Note concept of bounded sequence Above Below Bounded implies ________________ Both

Assignment Lesson 9.1 Page 602 Exercises 1 – 93 EOO

Series and Convergence Lesson 9.2

Definition of Series Consider summing the terms of an infinite sequence We often look at a _______________sum of n terms

Definition of Series We can also look at a _____________of partial sums { S n } The series can _________________________ The sequence of partial sums converges If the sequence { S n } does not converge, the series diverges and has no sum

Examples Convergent Divergent

Telescoping Series Consider the series Note how these could be regrouped and the end result As n gets large, the series = 1

Geometric Series Definition An infinite series The ______________of successive terms in the series is a ________________ Example What is r ?

Properties of Infinite Series ______________________ The series of a sum = the sum of the series Use the property

Geometric Series Theorem Given geometric series (with a ≠ 0) Series will Diverge when | r | _________ _______________when | r | < 1 Examples Compound interest Or

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? 20 cm

Assignment Lesson 9.2 Page 612 Exercises 1 – 69 EOO

The Integral Test; p-Series Lesson 9.3

Divergence Test Be careful not to confuse Sequence of general terms { a k } Sequence of partial sums { S k } We need the distinction for the divergence test If Then must _________ Note this only tells us about ______________. It says nothing about convergence

Convergence Criterion Given a series If { S k } is _________________________ Then the series converges Otherwise it diverges Note Often difficult to apply Not easy to determine { S k } is bounded above

The Integral Test Given a k = f(k) k = 1, 2, … f is positive, continuous, _____________for x ≥ 1 Then either both converge … or both _________________

Try It Out Given Does it converge or diverge? Consider

p-Series Definition A series of the form Where p is a _____________________ p-Series test Converges if _____________ ___________if 0 ≤ p ≤ 1

Try It Out Given series Use the p-series test to determine if it converges or diverges

Assignment Lesson 9.3 Page 620 Exercises odd

Comparison Tests Lesson 8.4

Direct Comparison Test Given If converges, then converges What if What could you conclude about these?

Try It on These Test for convergence, divergence Make comparisons with a geometric series or p-series

Limit Comparison Test Given a k > 0 and b k > 0 for all sufficiently large k … and … where L is finite and positive Then either both ___________… or both _________

Limit Comparison Test Strategy for evaluating 1.Find series with _______________ and general term "essentially same" 2.Verify that this limit exists and is positive 3.Now you know that _________________ as

Example of Limit Comparison Convergent or divergent? Find a p-series which is similar Consider Now apply the comparison

Assignment Lesson 9.4 Page 628 Exercises odd

Taylor and MacLaurin Series 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 Initial requirements M(c) = ____________ ____________ = f '(c) Centered at c or ____________

Linear Approximations The ____________________is a good approximation of f(x) for x near a a x f(a) f'(a) (x – a) (x – a) Approx. value of f(x) True value f(x)

Linear Approximations Taylor polynomial degree 1 Approximating f(x) for x near 0 Consider How close are these? f(.05) f(0.4)

Quadratic Approximations For a more accurate approximation to f(x) = cos x for x near 0 Use a __________________ function We determine At x = 0 we must have The functions to agree The first and ________________________ to agree

Quadratic Approximations Since We have

Quadratic Approximations So Now how close are these?

Taylor Polynomial Degree 2 In general we find the approximation of f(x) for x near 0 Try for a different function f(x) = sin(x) Let x = 0.3

Higher Degree Taylor Polynomial For approximating f(x) for x near 0 Note for f(x) = sin x, Taylor Polynomial of degree 7

Improved Approximating We can choose some other value for x, say x = c Then for f(x) = e x the n th degree Taylor polynomial at __________

Assignment Lesson 9.7A Page 656 Exercises 1 – 23 odd Lesson 9.7B Page 656 Exercises 25 – 43 odd