Section 9.2 Series and Convergence

Transcription

1 Sectio 9. Series ad Covergece

2 Goals of Chapter 9 Approximate Pi Prove ifiite series are aother importat applicatio of limits, derivatives, approximatio, slope, ad cocavity of fuctios. Fid challegig atiderivatives like e x dx Lay the groudwork for future courses

3 Summatio Notatio A compact otatio (ofte called sigma otatio) for sums is the followig: Upper Limit of Summatio i Geeral Term a a a a i Idex of Summatio Lower Limit of Summatio

4 Evaluate: 4i 3 i Examples 4 i= 4 i= 43 i= Series ivestigate the followig: i 4i

5 Sum all of the areas: Ifiite Sum What is the area of the square? = square uit Cut the square i half ad label the area of oe sectio. Cut the ulabeled area i half ad label the area of oe sectio. Cotiue the process Sice the ifiite sum represets the area of the square... The geeral term is

6 Ifiite Series A ifiite series is a expressio of the form a + a + a a +, or k= a k The umbers a, a, are the terms of the series; a is the th term.

7 Coectig Series ad Sequeces Cosider the Series: Fid the sum of The first term: s = 0.3 The first terms:s = = 0.33 The first 3 terms:s 3 = = The first 4 terms:s 4 = = Cosider the sequece of PARTIAL SUMS above: 0.3, 0.33, 0.333, The sequece of PARTIAL SUMS appear to coverge to: 3

8 Partial Sums of a Series The partial sums of the series for a sequece: s = a s = a + a s 3 = a + a + a 3 s = k= of real umbers, each defied as a fiite sum. a k

9 Coverget or Diverget Series If the sequece of partial sums has a limit S as, we say the series coverges to the sum S, ad we write a + a + a a + = k= a k = S Otherwise, we say the series diverges.

10 Examples Ivestigate the partial sums of the sequeces below to determie if the series coverges or diverges. If it coverges, state the limit.. = ( ) = = = ( ) 4. 0.() = = Diverges Diverges = 5 9 = = Diverges 5. = 00(0.5) = = = = Diverges Why do, 3, 4 ad 6 Diverge? The limit of the geeral term does ot equal 0.

11 The -th Term Test If lim a 0, the the ifiite series = a diverges. OR If the ifiite series = a coverges, the lim a = 0. Is the coverse of this statemet true? If lim a = 0, does the ifiite series = a always coverge? Whe determiig if a series coverges, always use this test first!

12 The Coverse of The -th Term Test Cosider the two famous sequeces below: Harmoic : Series 3 4 lim 0 Alteratig Harmoic Series 3 4 :... ( )... lim( ) 0 For both series, the lim a = 0. BUT do both series coverge? Check a calculator program.

13 The Alteratig Harmoic Series appears to coverge to ~0.69. The Harmoic Series appears to diverge. Harmoic Series The Coverse of The -th Term Test : Alteratig : ( ) Harmoic Series

14 The -th Term Test If lim a 0, the the ifiite series = a diverges. OR If the ifiite series = a coverges, the lim a = 0. If lim a = 0, the ifiite series Whe determiig if a series coverges, always use this test first! The coverse of this statemet is NOT true. = a does ot ecessarily coverge.

15 The Harmoic Series Diverges Prove the Harmoic Series diverges: = = Compare the Series to the graph of y = x. Fid the Left Had Riema Sum to approximate dx. x The Left Had Riema Sum is equal to the Sum of the Harmoic Series.

16 The Harmoic Series Diverges Prove the Harmoic Series diverges: = = Compare the Series to the graph of y = x. = So = The LHRS of x dx Sice y = is decreasig, the Left x Had Riema Sum is a over estimate. Thus: Left Had Riema Sum > x dx

17 The Harmoic Series Diverges Prove the Harmoic Series diverges: = = So = > Compare the Series to the dx x graph of y =. We ca fid the value of the x improper itegral: lim b x b dx b lim l x b lim l b b l Sice dx diverges ad x = > dx, the Harmoic x Series Diverges.

18 The Harmoic Series Diverges Part Justify that the Harmoic Series diverges aother way: = = Ivestigate the sum: By icreasig the size of, we ca make the sum of the ifiite series as large was we desire.

19 The Alteratig Harmoic Series Coverges Justify the Alteratig Harmoic Series coverges: = ( ) = Ivestigate ad plot the sum: Each Successive term i the sequece of partial sums is betwee the two previous terms i this sequece The sum is bouded by The sum must be betwee 0.5 ad. S? We will fid the actual value of the sum soo. ay two successive terms.

20 Arithmetic ad Geometric Series A Arithmetic Series has a costat differece betwee terms. (Similar to a Arithmetic Sequece.) Example: = ( 3) = A Geometric Series has a costat ratio betwee terms. (Similar to a Geometric Sequece.) Example: = 3(4) =

21 Arithmetic ad Geometric Series By the -th Term Test, every Arithmetic Series diverges: = (a + b) = a + b + a + b + + a + b lim (a + b) 0 Some Geometric Series diverge ad others coverge: = = = () 00(0.5) = = 5 9 = Diverges = = 00 Sice Geometric Series occasioally coverge, we will focus o them.

22 Defiitio of a Geometric Series I a geometric series each term is obtaied from its precedig term by multiplyig by the same umber r: a + ar + ar + ar ar + = = ar Examples: The previous examples are geometric. = 5 0 = 0.() = 00(0.5)

23 White Board Challege Fid the geeral term ad the sum of the first 0 terms of the sequece: a 8.5 s

24 Fiite Sum of a Geometric Series Fid the sum S of the first terms of a geometric series: Multiply by r. S = = ar = a + ar + ar + ar ar 3 S a ar ar ar... ar 3 rs ar ar ar... ar ar S rs a ar a r S r a r Check with the previous example. S r Subtract the two equatios. Solve for the sum. What happes to the sum as the value of icreases to ifiity?

25 Ifiite Sum of a Geometric Series Cosider : lim S = = ar Depeds o the value of r. = a + ar + ar + ar ar = lim a( r ) r if r if r S lim a r r Diverges S lim a r r 0 lim a r a r

26 Coverget Geometric Series The geometric series = ar coverges if ad oly if r <. If the series coverges, its sum is a. r Example: Fid the sum if it exists =. 5( ) 3. π + π 4 + π3 8 + Where a is the first term ad r is the costat ratio. a S r a 5.5 r S a /4 r Diverges /