Definition. Sequences. Sequence As A Function. Finding the n th Term. Converging Sequences. Divergent Sequences 12/13/2010. A of numbers. Lesson 9.

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1 Definition Sequences Lesson 9.1 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 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 1

2 Determining Convergence Manipulate algebraically Determining Convergence Use l'hôpital'srule to of the function conjugate expressions and take the limit Note we are relating limit of a sequencefrom the limit of a function Bounded, Monotonic Sequences 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 Bounded implies Both Definition of Series Series and Convergence Lesson 9.2 summing the terms of an infinite sequence We often look at a sum of n terms 2

3 Definition of Series Examples We can also look at a of partial sums { S n } Convergent The series can The sequence of partial sums converges If the sequence { S n } does notconverge, the series diverges and has no sum Divergent the series Telescoping 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 As ngets large, the series = 1 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 3

4 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 Lesson 9.2 Page 614 Exercises 1 17 EOO all EOO Divergence Test The Integral Test; p-series Lesson 9.3 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 onlytells us about. It says nothingabout convergence Given a series Convergence Criterion 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, fis positive, continuous, for x 1 Then both converge or both either 4

5 Try It Out Given Does it converge or diverge? p-series Definition A series of the form Where pis 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 Lesson 9.3 Page 622 Exercises 1 41 EOO all Direct Comparison Test Given Comparison Tests If converges, then converges Lesson 8.4 What if What could you conclude about these? 5

6 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 Example of Limit Comparison Convergent or divergent? Find a p-series which is similar 3. Now you know that as Now apply the comparison Lesson 9.4 Page 630 Exercises 5-33 EOO Ratio Test & Root Test Lesson 9.6 6

7 Ratio Test For a series of positive terms We realize that the sequence { a k } must be "rapidly" decreasing towards zero Use It or Lose It Use the ratio test for the following series Convergent or divergent? 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 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 However convergence does notguarantee Try this out with Lesson 9.6 Page 645 Exercises 5 10 all EOO Power Series Lesson 9.8 (Yes we re doing this before 9.7) 7

8 Definition A power series centered at 0 has the form Definition A power series centered at c has the form this as an extension of a polynomial in x this as an extension of a polynomial in x Examples Where are these centered? 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 < Rand diverges for x c > R 3. The series converges absolutely for all x Example the power series Dealing with Endpoints What happens at x = 0? Use generalized ratio test for x 0 Try this Converges trivially at x = 0 Use ratio test Limit = x converges when x < 1 Interval of convergence -1 < x < 1 8

9 Dealing with Endpoints Try Another Now what about when x = ±1? Again use ratio test At x = 1, diverges by the divergence test At x = -1, also diverges by divergence test Final conclusion, convergence set is (-1, 1) Should get which must be < 1 or -1 < x < 5 Now check the endpoints, -1 and 5 Power Lesson 9.8 Page 668 Exercises 1 33 EOO Taylor and MacLaurin Series Lesson 9.7 Taylor & Maclaurin Polynomials 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 cin its domain Initial requirements M(c) = = f '(c) Centered at cor Linear Approximations The is a good approximation of f(x)for x near a True value f(x) a (x a) x Approx. value of f(x) f'(a) (x a) f(a) 9

10 Linear Approximations Taylor polynomial degree 1 Approximating f(x) for x near 0 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 Since Quadratic Approximations So Quadratic Approximations We have Now how close are these? Taylor Polynomial Degree 2 In general we find the approximation of f(x) for x near 0 Higher Degree Taylor Polynomial For approximating f(x) for x near 0 Try for a different function f(x) = sin(x) Let x = 0.3 Note for f(x) = sin x, Taylor Polynomial of degree 7 10

11 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 Lesson 9.7A Page 658 Exercises 1 4 all 5-29 odd Remainder of a Taylor Polynomial Accuracy in Series We need a sense of how accurate our approximation is Lesson 9.7B Actual Approximate Remainder Function Value Remainder of a Taylor Polynomial Error Calculation Error associated with the approximation We can determine the maximum error with the formula Where Mis the bound on the n+1 st derivative of f(x) d is the number of good digits after the decimal 11

12 Error Calculation 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) We will be given f(x) from this we can determine M c the center Thus, given any two of x, n, and dyou can determine the other two Error Calculation Given f(x)we determine Mfor the interval [a, b] spanned by cand x Shortcuts If f(x) = sin(x) or cos(x), then M = 1 If f(x) = e x then If f(x) = e -x then Note signifies the "ceiling" function, the next integer beyond the largest value in the interval Note remaining shortcuts on handout Try It Out Fifth Maclaurin polynomial for sin x Determine P 3 (0.2) Use to determine the accuracy of the approximation Try It Out Some More Determine the degree of the Taylor Polynomial P n (x) expanded about c = 1 that should be used to approximate ln(1.3) so that the error is less than We are given d x We seek the value of n Note the interval is [1, 1.3] Lesson 9.7B Page 659 Exercises odd 12

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