Assignment 10 Mathematics 2(Model Answer)

Transcription

1 Assignment 0 Mathematics (Model Answer) Alternating Series / Absolute Convergence Problem: Determine whether of the series is convergent or divergent. ) (a) ( ) = (b) ( ) + = + ) (a) =7 ( ) (b) =4 ( ) 3 3) (a) = ( ) (b) 3 = + 4 4) (a) =4 ( ) (b) =4 ( ) ! 3 5) (a) =3 (b) =6 ( ) + 3e ) (a) = ( ) (b) =0 ( ) + ) a. ( ) = ln solution: This is an alternating series where: ) a = > 0 ) a = = 0 3) a + = a (+). = ( ) < a ++ +< a i.e., a is decreasing Or: using (calculus) f (x):f() =, f () = 4 3 < 0 f () is decreasing So from ), ) & 3) the series converges. = b) ( ) + solution: This is an alternating series where: ) a = + > 0

2 ) a + = use L Hopital = 0 So, the alternating series diverges. ) a. ( ) =7 3 solution: This is an alternating series where: )a = 3 > 0 ) 3 =, use l hopital = = 0 3 3)f() = 3, f () = 3 () ( )3 = 6+3 = 3 4 < f() is decreasing. from ), ) and 3) so, the alternating series converges 3 + b. ( ) =4 ) a = 3 > 0 ) 3 =, use l hopital 3 ln() = 0.So, the alternating series diverges. 3) a. ( ) + 4 solution: This is an alternating series where + ) a = 4 > 0

3 + ) a 4 =, use l hopital 4 ln 4 = = 0 3) f() = +, 4 f () = 4 () (+)4 ln 4 = (4 ) (ln 4 (+)) 4 < 0 So a is decreasing.from ), ) and 3) then the series converges. b. 3 = = 3 ( ) = this is a geaometric series where, < r = <.then it is convergent =4 4) a. ( ) + 3 Solution this is an alternating series where, ) a = + > ) =, use 3 l hopital = = 0 3 3) f() = +, 3 f () = 3 (+)3 = 3 < 6 4 0,a is decreasing From ), ) and 3).then series converges. b.! 3 =3 use ratio test So it is diverge a + ( + )! a 3 3 3! + 3 = > 3

4 4) a. 4 = solution: This is positive term series, use it comparison test with b = Where is convergent because this is a p-series with p= > a But b = 4 > 0 Then two series have same properties.so, =3 also converges b. ( ) 3e =6 This is an alternating series where, ) a = 3 e > 0 ) a = 3 e = 3 0.So, it diverges. 6) a. ( ) ln = This is an alternating series where, ) a = > 0 ln ) a = = 0 ln 3) f() = (ln ), f () = (ln ). = ( (ln ) < 0 It means a is decreasing.from ), ) and 3) then series converges. b. ( ) + =0 This series is an alternating series where, 4

5 ) a = > 0 ) = 0 So it diverges. Problem : Estimate the sum of each convergent series to within ) ( ) + = s-s a + = 3 (+) 0. 0( + ) ( + ) it means that a s-s 5 a = 00 = = = 5 so.sum~s 5 = a + a +a 3 +a 4 + a 5 = = ) ( ) =4 0 s-s a + = (+) 0 + Since a 3 < 0. 0 Then Sum~ a = > a = 4 > a 3 = 9 <

6 3) ( ) + =0 s-s a + =! (+)! 0. 0( + )! 0. 0 ( + )! 0. 0 = 00 = 5 it means that a 6 = = 0. 0 s-s 6! a 6 so.sum~s 5 = a + a a 3 +a 4 a 5 = + + = ) ( ) + 3 =3 s-s a + = 3 (+) 0. 0( + ) ( + ) = 300 = 6 it means that a 7 = 3 = 0. 0 s-s a 7 3 So.Sum~s 6 = a 3 a 4 +a 5 a 6 = = Problem 3: Determine how many terms are needed to estimate the sum of the series to within () ( ) =0! s-s a + = (+)! solution: a = > a = 4 >

7 a 3 = 8 > a 4 = 6 > a 5 = 3 > a 6 = 6 > ! a 7 = 7 > ! a 8 = 8 > ! a 9 = 9 > ! a 0 = 0 > ! a = < ! = 0 () ( ) = +! solution: s-s a + = (+)! (+) a = > a = (3)! (3) 3 = > a 3 = (4)! 3 = > (4) 4 3 a 4 = (5)! 4 = > (5) 5 65 a 5 = (6)! (6) 6 = 5 >

8 a 6 = (7)! (7) 7 = 70 > a 7 = (8)! (8) 8 = 35 > a 8 = (9)! (9) 9 = > a 9 = (0)! (0) 0 = > a 0 = ()! () = > a = ()! () = > = Problem 4: Determine whether the series is absolute convergent, conditionally convergent or divergent. ) (a) ( ) 6 =0 (b) ( )! =0 3 ) (a) ( ) + + = (b) =4 ( ) 0! 3) (a) ( ) + 4 =3 (b) ( ) 3 = 3 ) 4 + 4) (a) =5 ( (b) = 5) (a) = ( ) + (b) = 3 e 6) (a) ( e ) 7) (a) tan 8) (a) ( )+ 3 + =4 (b) = 4 cos = (b) ( ) = = (b) =8 3 ln 9) (a) ( ) + 4 =4 (b) ( ) 4 =0 0) (a) ( 3) 4 ) a. =0 ( ) 6!! 3 (+)! = (b) ( + = ) Solution 8

9 By using Ratio test. a + = a ( ) + 6 (+)!! ( ) 6 6! = (+)!(6) (+) = = 0 < so the series converges absolutely b. =( ) 3 By using Ratio test. a + a 3 3+ = = 3 = < 3 so the series converges absolutely ) a. =( ) + + To chec absolute convergent: By using Ratio test a + a = (+) = Test fail Use another test (Divergence Test) ( ) + + = + + = Using L Hopital s rule = + 3 = 0.So the series is not absolute convergent To chec conditionally convergent: )a = + > 0 + ) =, L Hopital s rule = 0.So series diverges. b. ( ) = 0! 9

10 To chec absolute convergent: By using Ratio test a + a 0+! (+)! 0 = = 0 < + 0 so the series converges absolutely ) a. =3( ) + 4 = + ( )+ 4 = 4 + =3 + By using Integral Test t t 3 + d ln + t so the series is not absolutely convergent To chec the conditionally convergent: t ln t + - ln7 =. a = 4 > a = f() = + f () = 8 < 0 (+) a is decreasing, So from, and 3 then the series is conditionally convergent b. ( ) 3 = To chec absolute convergent: By using Ratio test a + a (+) = 3 > So it is not absolutely convergent To Chec the conditionally convergent. a = 3 > 0. a = ( 3 ) So it is not conditionally convergent hence it is divergent =5 4) a. ( 3 4 ) To chec absolute convergent: By using Root, test 0

11 a ( 3 4 ) Use l Hopital Rule 3 = 3 < 4 4 So it is absolutely convergent 3 = 4 6) 4 = = 4 = since series harmonic then its (Divergent) ) a. ( ) + = 3 + To chec absolute convergent: ( ) + = = = Use L.C.T with b = P series and P = > So it is convergent a And b = 3 + = Use L Hopital s rule 3 3 = > 0 Then the two series have same properties series is absolutely convergent b. = 3 e By using Ratio test a + a (+)3 e.e. e = e 3 = e = e < So it is convergent ) a. ( e =4 ) This is a positive term series. Use Root Test ( e ) e = Use L Hopital s rule

12 e = = > so it is divergent cos b. = 3 This is not alternating series because the signs change irregularly cos 3 = = e cos = 3 Using comparison test cos = < 3 = 3 So the series, is absolutely convergent ) a. tan = This is a positive term test Use L.C.T with b = this is harmonic series (diverge) b a tan. tan = π > 0 tan The two series have same properties, so the = is also divergent ( ) b. = ln This is an alternating series To chec absolute convergent: ( ) = ln = ln f() = ln this is a Positive continuous function and decreasing,because f ()= 0 (. +ln ) = ( ln ) (+ln ) ( ln ) < 0 Then we can apply integral test ln (ln ) t d Then integration Diverges. t So the series is not absolutely convergent. t [ln (ln t)-ln (ln )]=

13 To chec conditionally convergent. a = > 0 ln. a = 0 3. f () < 0. a is a decreasing. Then series is conditionally convergent ) a. ( )+ This is an alternating Series To chec absolute convergent: = ( )+ = = P-Series and P= 0.5 < This is divergent, ( ) is not absolutely convergent To show that this series is conditionally converge or diverge. a = > 0. a = 0 3. f () < 0. a is a decreasing. Then series is conditionally convergent b. 3 This is a positive term series Use Ratio Test a a = < 3 So it is absolutely convergent ) a. ( ) + 4 =4 This is an alternating series To chec absolute convergent: Use Ratio test! (+)4 4! (+)! a + a Use l Hopital Rule 8 = 0<.So it is absolutely convergent = b. ( ) 4 =0 (+)! This is an alternating series To chec absolute convergent:use Ratio Test a + a 4.4 ((+)+)! (+)! 4 3

14 (+3)(+)(+)! 4 (+)! = 0 <.So it is absolutely convergent ) a. = ( 3) 4 = ( ) 3 = 4 This is an alternating series To chec absolute convergent: Use Ratio Test a + a 3.3 (+) = Use L Hopital s rule 8 =3 < 4 Therefore, it is absolutely convergent b. ( + ) This is a positive term series. Use Root Test ( + ) ( + ) = e > So it is divergent Problem 5: Name the method by identifying a test that will determine whether the series converges or diverges. ) = e ( + ) (Use the Root test) ) = (Use the L.C.T test) 3) ( 4 = 5 ) (Use the Geometric series test) = 4) ( ) = e! (Use the Ratio test) 5) 3 e (Use the Root test) =0 6) ( ) (Use the Alternating series test) 7) 8) = = 3 + cos π (Use the Integral test) (Use the Alternating series test) =3 9) ( ) ln( + ) (Use the Alternating series test) 0) 3 (Use the Ratio test = 4