( ) = A n + B ( ) + Bn

Transcription

1 MATH 080 Test 3-SOLUTIONS Fll 04. Determie if the series is coverget or diverget. If it is coverget, fid its sum.. (7 poits) = This is coverget geometric series where r = d <. The sum is r = / ( / ) = 3, i.e. ( ) = 3. b. (7 poits) ( +) + = A + B + B This is telescopig series. = A + = ( A + B) + A A = ; B = + Usig prtil frctio decompositio, = ( +) + S = S = + 3 S 3 = S = = + lim S + = The sequece of prtil sums coverges to. Thus, the telescopig series coverges to.

2 MATH 080 Test 3-SOLUTIONS Fll 04 c. (4 poits) Sice both series re coverget, the sum of the two series coverges. 5 ( +) ( ) 3 = = 5() ( +) ( ) 3 = 6. (6 poits) Cosider the series e. It c be show tht the series coverges by pplyig the Itegrl Test. Assume the coditios for the Itegrl Test hve bee verified. Use the Remider Estimte for the Itegrl Test to boud (lower d upper) the error whe the sum of the series is pproimted by ddig the first 7 terms. Let f = e d R = S S. The f d R 7 f d 8 f d b 7 b e d e b b b e e b = e Therefore, e 8 R 7 e 7.

3 MATH 080 Test 3-SOLUTIONS Fll 04 3 For problems 3-7, determie if the series coverges bsolutely, coverges coditiolly, or diverges (7 poits) The terms of the series lterte i sig.! ( ) 9! = 9! lim 9 + ( +)!!9 + 9 ( +)!9! = 0 <. 9 The series coverges by the Rtio Test. Therefore the series coverges!! bsolutely (7 poits) The terms of the series re positive Use the Limit Compriso Test. Compre the series hrmoic series, 5 4 lim b k to the diverget = < 5 3 < 5 4 The series diverges by the Limit Compriso Test

4 MATH 080 Test 3-SOLUTIONS Fll (7 poits) l + e The terms of the series re positive. lim l + e = l lim + e = l lim + e = l 0 By the Divergece Test, the series diverges. 6. (7 poits) cos + 3 All the terms of the series re positive. 8 0 cos 0 cos cos The series, with r = 8 < is coverget geometric series. Thus 4 is 8 8 coverget series. By the (Direct) Compriso Test, the series coverges. It coverges bsolutely. 7. (7 poits) = = The terms of the series re positive. This is coverget p- series with p = 3 >. The series coverges bsolutely. 3/

5 MATH 080 Test 3-SOLUTIONS Fll Cosider the series. The terms of the series lterte i sig. +. (0 poits) Use the Itegrl Test to show the series is ot bsolutely coverget. (Be sure to show verifictio tht the three coditios of the itegrl test re stisfied.) Cosider ( ) + = + + Let f () = +. Observe tht f () is positive d cotiuous for. To determie if f () is decresig, compute its derivtive. = + ( +) f () = + + The f () < 0 whe >. + ( +) < 0 whe + < 0 or > Thus f () is positive, cotiuous, d decresig for. The three coditios for the itegrl test re stisfied. Use the Itegrl Test to determie if the series coverges or diverges. d + b b + d ( ) b l = lim l b + l b + b = The improper itegrl diverges. By the Itegrl Test, the series diverges. Ad + the series +. Thus the series ( ) + does ot coverge bsolutely. = + + Note: Usig u- substitutio, let u = +, du = d. The if =,u = l5 d if = b, u = l( b +). + d = lim b b + 5 = lim b u du ( l( +) ) b 5 = lim l b + = b + l5 =

6 MATH 080 Test 3-SOLUTIONS Fll 04 6 b. (7 poits) Determie if the series, + Use the Altertig Series Test to determie if the series Note tht b =, is coditiolly coverget. + ( )+ + =. Two coditios must hold: + lim for ll Let f () = +. From prt ) + + ( ) + coverges. + < 0 whe + < 0 or >. Thus f () is decresig for > d 0 < b + b for, tht is the b re o- icresig. + By the Altertig Series Test, the series coverges. Sice it does ot + coverge bsolutely, it coverges coditiolly. c. (4 poits) If the series is coditiolly coverget, use the Altertig Series Estimtio Theorem to provide boud o the bsolute vlue of the error, R 4, whe the sum of the series is pproimted by ddig up the first 4 terms. Add up the first 4 terms to pproimte the sum of the series. The boud o the bsolute vlue of the error i this pproimtio is the et term i the series, i.e. R 4 5. Thus R (6 poits) Suppose the series coverges. Does the sequece { }coverge or diverge. If it coverges, the to wht vlue does it coverge? Justify your swer with cler, cocise sttemet. The sequece coverges to 0. If series coverges, the sequece of terms coverges to zero.

7 MATH 080 Test 3-SOLUTIONS Fll Cosider the series. 4. (7 poits) Alyticlly determie the rdius of covergece, R, for the series. Set up d simplify the rtio: ( +) + + ( +)4 + = = + ( +) + 4 Evlute the limit: lim ( +) = + 4 lim = = + + = 4 4 Apply the Rtio Test: The power series coverges whe lim + + <, i.e. < or + < 4. Thus the 4 rdius of covergece for the power series is R = 4. b. (7 poits) Fid the itervl of covergece for the series. Sice + < 4, the 4< + < 4 5< < 3 So the iterior of the itervl of covergece is 5 < < 3. Checkig for covergece t the edpoits of the itervl: If = 5, the = = = = 4 = 4 = 4 = the coverget ltertig hrmoic series.. The series ( ) is 3+ 4 If = 3, the = = =. The series 4 4 hrmoic series. is the diverget The complete itervl of covergece for the power series is 5 < 3.