Testing for Convergence

Transcription

1 9.5 Testig for Covergece

2 Remember: The Ratio Test: lim + If a is a series with positive terms ad the: The series coverges if L <. The series diverges if L >. The test is icoclusive if L =. a a = L

3 This sectio i the book presets several other tests or techiques to test for covergece, ad discusses some specific coverget ad diverget series.

4 Nth Root Test: lim a If a is a series with positive terms ad = L the: The series coverges if L <. The series diverges if L >. The test is icoclusive if L =. Note that the rules are the same as for the Ratio Test.

5 example: = = lim = ( lim )?

6 lim lim lim l e ( ) formula #04 lim e 0 e lim l e l lim e formula #03 Idetermiate, so we use L Hôpital s Rule

7 example: = = lim lim = ( lim ) = =? = it coverges

8 aother example: = = lim = = it diverges

9 Remember that whe we first studied itegrals, we used a summatio of rectagles to approximate the area uder a curve: 3 This leads to: a { } The Itegral Test f ( ) If is a positive sequece ad where f ( ) is a cotiuous, positive, decreasig fuctio, the: a = a ( x) = N ad both coverge or both diverge. N f dx

10 Example # = Does coverge? x x dx = lim b b x 3 dx = lim b b x = lim + b b = Sice the itegral coverges, the series must coverge. (but ot ecessarily to.)

11 Example # Apply the itegral test to the series: + = x Let f ( x) = x + the, check for 3 coditios:

12 Example # Apply the itegral test to the series: + =. f(x) positive. f(x) cotiuous 3. f(x) decreasig

13 Example # Apply the itegral test to the series: + = Itegrate: x x + dx

14 Example # Apply the itegral test to the series: + = Coclusio?

15 Apply the itegral test: Hit: dx = arcta x x + + = Try This The itegral coverges covergig to π / 4 does t mea the series coverges to the same value.

16 Problems with the Itegral Test: You must show that fis positive, cotiuous ad decreasig for x. You must be able to fid the ati-derivative.

17 p-series Test = = p p p p 3 p > p coverges if, diverges if. We could show this with the itegral test. If this test seems backward after the ratio ad th root tests, remember that larger values of p would make the deomiators icrease faster ad the terms decrease faster.

18 Try This Does the followig series coverge or diverge? Justify your aswer. = coverges It is a p-series with p>

19 the harmoic series: = = diverges. (It is a p-series with p=.) It diverges very slowly, but it diverges. Because the p-series is so easy to evaluate, we use it to compare to other series.

20 Limit Compariso Test a > 0 b > 0 N If ad for all (N a positive iteger) a lim = c 0 < c < a b If, the both ad a lim 0 b b coverge or both diverge. If, the coverges if coverges. a = b a lim b If, the diverges if diverges. = a b

21 Importat Ideas Use the Limit Compariso Test to compare a series with ukow behavior with a series with kow behavior. Ofte you ca select a p- series for compariso. Whe selectig a p-series, use the same degree umerator ad deomiator as the give series ad disregard all but the highest powers of.

22 Example 3a: = ( ) = + Whe is large, the fuctio behaves like: a > lim b + ( ) = lim + = lim = ( + ) ( + ) = lim = harmoic series Sice diverges, the series diverges.

23 Example 3b: = = Whe is large, the fuctio behaves like: a lim b lim = = lim = geometric series Sice coverges, the series coverges.

24 Stop here for today Assigmet p53,,5-5 odd

25 Alteratig Series Good ews! The sigs of the terms alterate. Alteratig Series Test If the absolute values of the terms approach zero, the a alteratig series will always coverge! example: + ( ) = = This series coverges (by the Alteratig Series Test.) This series is coverget, but ot absolutely coverget. Therefore we say that it is coditioally coverget.

26 Example: Determie the divergece or covergece of: a Is a a +? Does lim a 0? = ( ) = +

27 = =3 =5=7 { + ( ) = = =4=6=8 Is the alteratig sequece covergig?

28 Try This Determie the covergece or divergece of = ( ) + + Diverges

29 Try This = ( ) + + fails the alteratig series test for covergece. Does this mea it diverges? Yes How ca you prove that it diverges? -th term test

30 Sice each term of a coverget alteratig series moves the partial sum a little closer to the limit: Alteratig Series Estimatio Theorem For a coverget alteratig series, the trucatio error is less tha the first missig term, ad is the same sig as that term. This is a good tool to remember, because it is easier tha the LaGrage Error Boud.

31 Example: Approximate the sum of the followig coverget, alteratig series by evaluatig the first 3 partial sums. ( ) +! = Estimate the error.

32 Example S = S = S = + 3 = ( ) ! Error < { a } = = ! 4

33 ENDPOINT TESTING For a power series whose radius of covergece is a fiite R, there are edpoits which may be either coverget or diverget. c R R coverget ( coverget ) diverget

34 For a power series whose radius of covergece is a fiite R, there are edpoits which may be either coverget or diverget. R c R coverget [ coverget coverget ) diverget

35 For a power series whose radius of covergece is a fiite R, there are edpoits which may be either coverget or diverget. R c R coverget ( coverget ] diverget coverget

36 For a power series whose radius of covergece is a fiite R, there are edpoits which may be either coverget or diverget. c R R coverget [ coverget ] coverget

37 Fid the iterval of covergece = x Hit: fid the radius of covergece usig the Ratio Test the test the edpoits

38 Try This Fid the iterval of covergece. Be sure to test edpoits. = x [-,]

39 Assigmet : p53:7-0 all, 35a, 37a, 4,a