Review (11.1) 1. A sequence is an infinite list of numbers {a n } n=1 = a 1, a 2, a 3, The sequence is said to converge if lim

Transcription

1 Announcements: Note that we have taking the sections of Chapter, out of order, doing section. first, and then the rest. Section. is motivation for the rest of the chapter. Do the homework questions from section. and.,.2 and.3. Try the questions from section.,. and.2 and.3 on your study guide. Read over the Maple project and download the associated Maple worksheets and make certain they work on your computer. Practice the skills questions from 3.2 and 4.3 (Quiz 8a) and from 4.3, 2.3, 2.4 (Quiz 7b) for Wednesday.

2 2 Review (.). A sequence is an infinite list of numbers {a n } = a, a 2, a 3, The sequence is said to converge if lim n a n exists and is finite. 3. A sequence is said to diverge if it does not converge. 4. A sequence is said to be bounded if there is one constant M so a n M for all terms a n in the sequence. 5. A sequence is said to be monotonic if it is either increasing or decreasing (a) It is decreasing if a n a n+ for all n =, 2, 3,... (b) It is increasing if a n a n+ for all n =, 2, 3,... Try the following problems. lim n n! (n + 2)! 2n + n 2 2. lim n 3n 2 + n 3. Is { 2n 3n+} monotonic? 4. Is {( ) n n} bounded?

3 3 4 n Question What is lim n n! A. 4 B. C. 0 D. 4 E. None of the above

4 4 n. Answer to Question What is lim n n! 4 A. 4 B. C. 0 is the correct answer. D. 4 E. None of the above

5 5 Series (.2) Series. A series is a sum of infinitely many terms a k = a + a 2 + a k= 2. The partial sums of a series are sums that stop after n terms n S n = a k = a + a 2 + a a n k= 3. A series is said to converge if lim S n = lim a + a 2 + a a n n n exists and is finite. If a series converges, the terms add up, if not it is said to diverge. 4. Whether a series converges is a tricky problem for which there are many different convergence tests.

6 6 The Geometric Series. The geometric series is a series of the form ax k. (a 0) 2. The geometric series is special in that we can find a formula for the partial sums S n = n k=0 k=0 ax k = a xn+, (x ) x For most other series, we cannot find such a simple formula. 3. Based on the formula for the partial sums we have If x < then ax k = a. (The series converges) x k=0 If x then ax k diverges. k=0 Questions:. Prove parts 2 and 3 for the geometric series. 2. For which x does 2 n+ x n converge? To what does it converge

7 7 Telescoping Series (.2) Another series that is easy to analyze is the telescoping series. They are called this because the partial sum collapse like a telescope. It is easiest to analyze these by writing out terms, so we look at examples. By collapsing the partial sums, show the following (. n ) = n + 2. (ln(n + ) ln(n)) diverges.

8 Question The series A. converges to e 3 ( e (n+3) e (n+2)) 8 B. converges to e 3 C. converges but we can t say to what D. diverges E. none of the above

9 Answer to Question The series ( e (n+3) e (n+2)) 9 A. converges to e 3 B. converges to e 3 is the correct answer. C. converges but we can t say to what D. diverges E. none of the above

10 0 Harmonic Series (.2) The harmonic series is n = Preliminaries. The terms of this series are a =, a 2 = 2, a 3 = 3, a 4 = 4, The limit of the terms in the series is zero. 3. The Partial sums of this series are S =, S 2 = + 2, S 3 = , S 4 = , and so on 4 4. The Partial sums are much more difficult to analyze than the terms, but it is the partial sums that will determine convergence or divergence. Results. Show that S 2 = + /2 S 4 + 2/2 S 8 + 3/2 S 6 + 4/2 S 2 n + n/2 and hence the partial sums approach infinity, so the series diverges, but very slowly. 2. The harmonic series diverges even though the terms go to zero. 3. Thus, even if we keep adding on smaller terms each time, we may not get the series to converge. 4. For series to converge, the terms must approach zero fast enough. 5. Determining what fast enough means is a difficult problem and the reason for all the convergence tests.

11 Divergence Test (.2) Consider a series a n Preliminaries. The terms of a series are a, a 2, a 3, a 4, The Partial sums of this series are S, S 2, S 3, S 4, The Partial sums are much more difficult to analyze than the terms, but it is the partial sums that will determine convergence or divergence. Results:. If the series converges then the partial sums approach a finite limit, and If the partial sums approach a finite limit, then the series converges. (This is the definition of convergence) 2. If the series converges then the terms must go to zero. 3. If the terms do not go to zero then the series diverges. (Contrapositive of statement 2) The divergence test is statement 3: If the terms of series do not go to zero then the series diverges.

12 2 Divergence Test (.2) The divergence test is: If the terms of a series do not go to zero then the series diverges. Caution The logic is filled with one way streets.. If the terms go to zero then the series may or may not converge, we don t know without further analysis. Examples. Does n 2 converge? 2. Does 3. Does ( ) n converge? n=2 n ln(n) converge?

13 3 Question Which of the following is false. A. The harmonic series diverges even though the terms approach zero B. If the partial sums of a series approach the number π, then the series converges C. If the terms of a series do not approach zero, then the series diverges D. Whenever the terms of a series approach zero, then the series must converge. E. Whenever the partial sums of a series tend to zero, then the series converges

14 Answer to Question Which of the following is false. 4 A. The harmonic series diverges even though the terms approach zero B. If the partial sums of a series approach the number π, then the series converges C. If the terms of a series do not approach zero, then the series diverges D. Whenever the terms of a series approach zero, then the series must converge. is the correct answer. E. Whenever the partial sums of a series tend to zero, then the series converges

15 5 Integral Test (.3) This is one of many tests for convergence. It relates the convergence or divergence of series to the convergence or divergence of improper integrals. It applies to series of positive terms when we can integrate the corresponding function. The Integral Test For the series Conditions. If a n 0 for all n and 2. If f(x) is a continuous positive function for all x so that f(n) = a n for integers n =, 2, 3,... and 3. The function f(x) is decreasing then Results. If the integral converges. 2. If the integral diverges. a n f(x) dx converges then the series f(x) dx diverges then the series a n a n Questions. Does n ln(n) 2. Does n=2 converge or diverge? converge or diverge? n2 3. Explain why the integral test works and illustrate with a picture.

16 Question For the series n=2 A. The series converges to n(ln(n)) 2 ln(2) B. The series converges, but we don t know the value to which it converges. C. The series diverges. D. The series neither converges no diverges. E. None of the above 6

17 Answer to Question For the series n=2 n(ln(n)) 2 7 A. The series converges to ln(2) B. The series converges, but we don t know the value to which it converges. is the correct answer. C. The series diverges. D. The series neither converges no diverges. E. None of the above