Absolute Convergence and the Ratio Test

Transcription

1 Absolute Convergence and the Ratio Test MATH 211, Calculus II J. Robert Buchanan Department of Mathematics Spring 2018

2 Bacground Remar: All previously covered tests for convergence/divergence apply only to positive term series (except for the Alternating Series Test).

3 Bacground Remar: All previously covered tests for convergence/divergence apply only to positive term series (except for the Alternating Series Test). Question: what about series which do not consist exclusively of positive terms, but which are not alternating series?

4 Bacground Remar: All previously covered tests for convergence/divergence apply only to positive term series (except for the Alternating Series Test). Question: what about series which do not consist exclusively of positive terms, but which are not alternating series? Example cos 2 = cos 1 + cos cos

5 Absolute Convergence Definition An infinite series a is absolutely convergent if the series converges. a = a 1 + a 2 + a 3 +

6 Absolute Convergence Definition An infinite series a is absolutely convergent if the series converges. a = a 1 + a 2 + a 3 + Remar: The series a is a positive term series.

7 Conditional Convergence Definition An infinite series a is conditionally convergent if the series converges but the series a diverges.

8 Conditional Convergence Definition An infinite series a is conditionally convergent if the series converges but the series For an arbitrary series a diverges. a, the series may be classified in only one of the following ways: absolutely convergent conditionally convergent divergent

9 Examples Determine which of the following infinite series are absolutely convergent, conditionally convergent, or divergent. cos ( 1) +1 1 ( 1) tan 1 3 sin( π/6)

10 cos 2 cos 2 1 for all = 1, 2,.... 2

11 cos 2 cos 2 1 for all = 1, 2, The series converges (p-series Test) which implies 2 cos 2 converges.

12 cos 2 cos 2 1 for all = 1, 2, The series converges (p-series Test) which implies 2 Therefore, cos 2 cos 2 converges. converges absolutely.

13 ( 1) ( 1)+1 1 for all = 1, 2,....

14 ( 1) ( 1)+1 1 for all = 1, 2, The series diverges (harmonic series).

15 ( 1) ( 1)+1 1 for all = 1, 2, The series diverges (harmonic series). The series ( 1) +1 1 converges by the Alternating Series Test.

16 ( 1) ( 1)+1 1 for all = 1, 2, The series diverges (harmonic series). The series ( 1) +1 1 converges by the Alternating Series Test. Therefore, ( 1) +1 1 converges conditionally.

17 ( 1) tan 1 3 ( 1) tan 1 3 π/2 for all = 1, 2,.... 3

18 ( 1) tan 1 3 ( 1) tan 1 3 π/2 for all = 1, 2, The series converges (p-series Test) which implies 3 π/2 3 converges.

19 ( 1) tan 1 3 ( 1) tan 1 3 π/2 for all = 1, 2, The series converges (p-series Test) which implies 3 π/2 3 converges. By the Comparison Test ( 1) tan 1 3 converges.

20 ( 1) tan 1 3 ( 1) tan 1 3 π/2 for all = 1, 2, The series converges (p-series Test) which implies 3 By the Comparison Test Therefore, π/2 3 ( 1) tan 1 3 converges. ( 1) tan 1 3 converges. converges absolutely.

21 sin( π/6) sin( π/6) 1 for all = 1, 2,.... 3/2

22 sin( π/6) sin( π/6) 1 for all = 1, 2,.... 3/2 1 The series converges (p-series Test) which 3/2 implies sin( π/6) converges.

23 sin( π/6) sin( π/6) 1 for all = 1, 2,.... 3/2 1 The series converges (p-series Test) which 3/2 implies sin( π/6) converges. Therefore, sin( π/6) converges absolutely.

24 Absolute Convergence Implies Convergence Theorem If a converges then a converges.

25 Absolute Convergence Implies Convergence Theorem If a converges then a converges. Proof. a a a 0 a + a 2 a Therefore (a + a ) converges by the Comparison Test. (a + a a ) = a converges.

26 Ratio Test Theorem (Ratio Test) Given a, with a 0 for all, suppose that Then lim a +1 a = L. 1. if L < 1, the series converges absolutely, 2. if L > 1, the series diverges, 3. if L = 1, there is no conclusion.

27 Examples Use the Ratio Test to determine the convergence or divergence of the following series ! 1 1 2

28 20 2 lim (+1) = lim = lim 2 ( + 1 ( ) 20 ) 20 = 1 2 < 1

29 20 2 lim (+1) = lim = lim 2 ( + 1 ( ) 20 ) 20 = 1 2 < converges absolutely.

30 ! lim (+1) +1 (+1)!!! ( + 1) +1 = lim ( + 1)! = lim = lim 1 ( + 1) ( + 1) ( = lim ) = e > 1

31 ! lim (+1) +1 (+1)!!! ( + 1) +1 = lim ( + 1)! = lim = lim 1 ( + 1) ( + 1) ( = lim ) = e > 1! diverges.

32 1 lim = lim + 1 = 1 The Ratio Test reaches no conclusion. The series is recognized as the harmonic series and therefore diverges.

33 1 2 lim 1 (+1) = lim 2 ( + 1) 2 = 1 The Ratio Test reaches no conclusion. The series is recognized as a convergent p-series and therefore converges absolutely.

34 Root Test Theorem (Root Test) Given a, suppose that Then lim a = L. 1. if L < 1, the series converges absolutely, 2. if L > 1, the series diverges, 3. if L = 1, there is no conclusion.

35 Examples Use the Root Test to determine the convergence or divergence of the following series. (ln ) / = (ln ) 2 2

36 (ln ) /2 lim (ln ) /2 (ln ) = lim /2 = lim (ln ) 1/2 = 0 < 1

37 (ln ) /2 lim (ln ) /2 (ln ) = lim /2 (ln ) 1/2 = lim = 0 < 1 (ln ) /2 converges absolutely.

38 2 +1 (ln ) =2 lim 2 +1 (ln ) = lim 2 +1 (ln ) = lim 2 1+1/ ln = 0 < 1

39 2 +1 (ln ) =2 lim 2 +1 (ln ) = lim 2 +1 (ln ) 2 1+1/ = lim ln = 0 < 1 = (ln ) converges absolutely.

40 2 2 lim 2 2 = lim 2 2 = lim 2/ 2 = 1

41 2 2 lim 2 2 The Root Test is inconclusive. according to the Ratio Test. = lim 2 2 2/ = lim 2 = converges absolutely,

42 Homewor Read Section 8.5 Exercises: 1 43 odd