BC: Q401.CH9A Convergent and Divergent Series (LESSON 1)

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1 BC: Q40.CH9A Coverget ad Diverget Series (LESSON ) INTRODUCTION Sequece Notatio: a, a 3, a,, a, Defiitio: A sequece is a fuctio f whose domai is the set of positive itegers. Defiitio: A ifiite series (or simply a series) is a expressio of the form a a a a k. k Each umber ak is a term of the series, ad a is the th term.

2 POSITIVE TERM SERIES: Observatios Test for Covergece or Divergece Theorem: th-term test (i) If lim a 0, the the series a is diverget. (ii) If lim a 0, the further ivestigatio is ecessary to determie whether the series a is coverget or diverget. Illustratio: Series th-term Test Coclusio Diverges, by tt lim 0 lim 0 Further ivestigatio is ecessary, by tt Further ivestigatio is lim 0 ecessary, by tt e e Diverges, by tt lim Theorem: Geometric Series Test Let a 0. The geometric series ar a ar ar ar (i) coverges ad thus has a sum (ii) diverges if r a S if r r Defiitio: A p-series is a series of the form, where p is a positive real umber. p p p 3 p Theorem: p-series Test (i) coverges if p (ii) diverges if p p

3 POSITIVE TERM SERIES: Formal Tests for Covergece or Divergece (These tests will ot give the sum S of the series, but istead will tell us whether a sum exists) INTEGRAL TEST for covergece (Lesso) If a is a series, let f ( ) a ad let f be a fuctio obtaied by replacig with x. If f is a positive-valued, cotiuous, ad decreasig for every real umber x, the the series a coverges if f ( x) dx coverges diverges if f ( x) dx diverges DIRECT COMPARISON TEST (Basic Compariso Test) for covergece (Lesso) Let a ad b be positive-term series. If b coverges ad If b diverges ad a b a b for every positive iteger, the a coverges. for every positive iteger, the a diverges. LIMIT COMPARISON TEST for covergece (Lesso) Let a ad b be positive-term series. If there is a positive real umber c such that a lim c 0, the either both series coverge or both series diverge. b RATIO TEST for coverges (Lesso) a Let a be a positive-term series, ad suppose that lim L. a If L <, the series is coverget. If L >, or lim a a, the series is diverget. If L =, apply a differet test; the series may be coverget or diverget. a Deletig terms of least Choice of b magitude

4 NOTES I: Determie by observatio whether the followig series coverge or diverge. Justify your aswer. A. 0. B. 5 C. D. E. F.

5 Notes II.. Determie whether the harmoic series coverges or diverges. 3 : Determie whether the ifiite series e coverges or diverges.

6 3: Determie whether the series 5 coverges or diverges usig the DCT 4: Determie whether the series 3 coverges or diverges usig the DCT

7 5: Determie whether the series 3 coverges or diverges usig the LCT. 6: Determie whether the series Let a coverges or diverges usig the LCT.

8 Lesso - Homework Formal Testig. Use the Itegral Test to determie if 5 coverges or diverges. Pg. 53 #7. Use the Itegral Test to determie if coverges or diverges. Pg. 53 #0 3. Use the DCT to determie if l coverges or diverges. Pg. 53 # Use the DCT to determie if coverges or diverges. Pg. 53 #5 5. Use the LCT to determie if coverges or diverges. Pg. 53 #6 Observatioal Testig: th term test/p-series/geometric series 6. Determie if coverges or diverges. Justify. Pg. 5 #9 7. Determie if coverges or diverges. Justify. Pg. 5 # Determie if coverges or diverges. Justify. Pg. 5 #38 9. Determie if () coverges or diverges. Justify. Pg. 5 # Determie if 3 coverges or diverges. Justify. Pg. 53 #8. Determie if l coverges or diverges. Justify. Pg. 53 #. Determie if 5 coverges or diverges. Justify. Pg. 53 #7 (yes you did this already)

9 BC: Q40.CH9A Coverget ad Diverget Series (LESSON ) POSITIVE TERM SERIES: Formal Tests for Covergece or Divergece INTEGRAL TEST for covergece (Lesso) If a is a series, let f ( ) a ad let f be a fuctio obtaied by replacig with x. If f is a positive-valued, cotiuous, ad decreasig for every real umber x, the the series a coverges if f ( x) dx coverges diverges if f ( x) dx diverges DIRECT COMPARISON TEST (Basic Compariso Test) for covergece (Lesso) Let a ad b be positive-term series. If b coverges ad a If b diverges ad a b b for every positive iteger, the a coverges. for every positive iteger, the a diverges. LIMIT COMPARISON TEST for covergece (Lesso) Let a ad b be positive-term series. If there is a positive real umber c such that a lim c 0, the either both series coverge or both series diverge. b RATIO TEST for coverges (Lesso) a Let a be a positive-term series, ad suppose that lim L. a If L <, the series is coverget. If L >, or lim a a, the series is diverget. If L =, apply a differet test; the series may be coverget or diverget. ROOT TEST for covergece (Not o the AP outlie) Let a ad b be positive-term series, ad suppose that lim a L. If L <, the series is coverget. If L >, or lim a, the series is diverget. If L =, apply a differet test; the series may be coverget or diverget.

10 I. Positive Term: Ratio ad Root Test Practice : Determie whether the ifiite series 3 coverges or diverges usig the Ratio Test.! : Determie whether the ifiite series 3 coverges or diverges usig the Ratio Test.

11 3: Determie whether the ifiite series coverges or diverges usig the Ratio Test.! 4: Determie whether the ifiite series 3 coverges or diverges usig the Root Test.

12 II. POSITIVE TERM SERIES: Formal Tests for Covergece or Divergece DIAGRAM

13 III. POWER (geometric) SERIES: For what values of x does the series coverge? A. Fid the iterval of covergece (the values of x for which the series coverges). Justify: : 0 x 0 : x 3: x 3 0 4: 0 x 9

14 B. Write a ifiite power series to represet the fuctio f (x) ad determie for what values of x the ifiite power series is true. 4. f ( x) x 5. 5 f ( x) 3x

15 Lesso - Homework Formal Testig Positive Term Series (Ratio Test). Determie whether the ifiite series Pg. 5 #35 0 e coverges or diverges usig the Ratio Test.. Determie whether the ifiite series Pg. 5 #37 3! coverges or diverges usig the Ratio Test. 3!! 3 3. Determie whether the ifiite series Pg. 5 #40 0! e coverges or diverges usig the Ratio Test. 4. Determie whether the ifiite series Pg. 5 #43!! coverges or diverges usig the Ratio Test. 5. Determie whether the ifiite series Pg. 5 #44 Power Series (Geometric)! coverges or diverges usig the Ratio Test. Fid the iterval of covergece of the series ad, withi this iterval, the sum of the series as a fuctio of x x 4 7. l 8. 0 x : Pg. 5 #3 x : Pg. 5 #6 3 : Pg. 5 #7 9. si x : Pg. 5 #8 0

16 BC: Q40.CH9A Coverget ad Diverget Series (LESSON 3) NON-POSITIVE TERM SERIES: Theorems / Tests for Covergece or Divergece CAB THEOREM THE CONVERGENCE IN ABSOLUTE (CAB) THEOREM If a coverges, the a. Note: If a diverges, the a may or may ot coverge ALTERNATING SERIES TEST (AST) (Observatioal Test) FORMAL ALTERNATING SERIES TEST : SEE PAGE 57 ALTERNATING SERIES TEST If the terms of the series (i) strictly alterate ad (ii) decrease i absolute value to zero, the the series coverges ALTERNATING SERIES TEST as the Glorified th Term Test If If a a ad lim 0 a ad lim 0 a, the a, the a coverges. a diverges as it fails the th term test. We see that this series is strictly alteratig because of the alteratig idicator expressio: The series will icrease i absolute value to zero if it passes the th term test: lim a 0 What do we show: We show lim a 0 What do we say: We say The series strictly alterates ad decreases i absolute value to zero What do we coclude: We coclude Therefore the series a coverges by the A.S.T

17 I. No Positive Term: The Coverge i Absolute (CAB) Theorem : Determie whether the ifiite series coverges or diverges. : Determie whether the ifiite series coverges or diverges

18 3: Determie whether the ifiite series: si si 3 si si 3 coverges or diverges. 4: Determie whether the ifiite series 4 coverges or diverges.

19 II. No Positive Term: The Alteratig Series Test (AST) : Determie whether the ifiite series coverges or diverges Determie whether the ifiite series coverges or diverges Determie whether the ifiite series coverges or diverges.

20 III. Coverget Series: Absolute ad Coditioal Covergece

21 IV. POWER SERIES (No-Geometric): INTERVAL OF CONVERGENCE. Fid the iterval of covergece of the power series Also state the ceter ad radius of covergece. x.

22 . Fid the iterval of covergece of the power series Also state the ceter ad radius of covergece. x 3 ( ) 0.

23 Lesso 3 - Homework IV. Power Series (No - Geometric) Fid the iterval of covergece of the power series. Also state the ceter ad radius of covergece.. x : Pg. 53 #4 3 x. : Pg. 53 # x! 4.! x 4 : Pg. 53 #4 : Pg. 53 #46 II. Covergece Testig No Positive Term Series 5. Determie whether the ifiite series coverges or diverges. l Pg. 53 #5 6. Determie whether the ifiite series Pg. 53 #9 coverges or diverges. 7. Determie whether the ifiite series Pg. 53 #7 8. Determie whether the ifiite series Pg. 53 #3! coverges or diverges. cos( ) coverges or diverges. III. Absolute Covergece, Coditioal Covergece, Divergece ( 9 ). Classify each series above (5 8) as absolutely coverget, coditioally coverget, or diverget. Show all work.

24 BC: Q40.CH9A Coverget ad Diverget Series (LESSON 4) **GRAND FINALE REVIEW** Fid the iterval of covergece of the power series. 0. ( ) 4 x : Pg. 53 #37 Fid the iterval of covergece of the power series. Also state the ceter ad radius of covergece x x 3 : Pg. 53 #43 3. x : Pg. 53 #47 : Pg. 53 #48 Fid the iterval of covergece of the series ad, withi this iterval, the sum of the series as a fuctio of x x l : Pg. 53 #39 x : Pg. 53 #50 Write the series as a polyomial of order =,, 3, 4, 5, ad 6. Illustrate this graphically (Class Illustratio) x x

25 BC: Q40.CH9A Coverget ad Diverget Series (Test Topics) I. POSITIVE TERM SERIES (Does the series coverge or diverge?) A. Observatioal Tests. th term test. geometric series 3. p-series 4. The Harmoic Series B. Formal Test. Itegral Test. Direct Compariso Test 3. Limit Compariso Test 4. Ratio Test 5. Root Test (Not o AP outlie) II. NON-POSITIVE TERM SERIES (Does the series coverge or diverge?) A. Observatioal Test. Alteratig Series Test. The Alteratig Harmoic Series B. Formal Test. CAB Theorem III. POWER SERIES (For what values of x does the series coverge?) A. Geometric ad Sum B. No-Geometric with edpoit testig

26 **Positive Term Series Review ** Determie whether each positive-term-series coverges or diverges. There may be more tha oe test that yields coclusive results e ! 8.

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