MAT 182: Calculus II Test on Chapter 9: Sequences and Infinite Series Take-Home Portion Solutions

Transcription

1 MAT 8: Clculus II Tst o Chptr 9: qucs d Ifiit ris T-Hom Portio olutios. l l l l 0 0 L'Hôpitl's Rul 0

2 . Bgi by computig svrl prtil sums to dvlop pttr: Th squc of prtil sums is s follows:,,,,, 7 9 Nxt, lyz th squc of prtil sums: Numrtors: itgrs Domitors: odd itgrs,,,,

3 . 7 As, 7 From th prcdig lysis, logicl choic for compriso sris is. Not tht, which is sclr multipl of th divrgt sris. clr multiplictio dos ot ffct covrgc, so th compriso sris is lso divrgt. A pproprit lgbric compriso btw th giv d compriso sris must b stblishd: Thus, 7, mig ch trm of th giv sris corrspodig trm i th compriso sris is grtr th th 7, which divrgs. By th Compriso Tst, 7 divrgs.

4 . ; 0 Bgi by computig svrl trms to dvlop pttr: Th squc is s follows: 0,, 6,, 0, 6, Nxt, lyz th squc by comprig trms to, ,,,, tc., to srch for pttr: Th diffrc btw corrspodig tris i ch row of th highlightd colums bov is.

5 R b dx dx 0 0 b x x u-substitutio: u x x u du dx x b u b R b 0 b u du 0 b b u du u 0 b b 0 b u b 0 b b 0 0 b 0

6 . (cotiud) b. Th rmidr R must b o grtr th 0 : , , , so th miimum umbr of trms rquird is 6. c. Lowr boud: Uppr boud: L dx 0 x U dx 0 x Us dirct substitutio ito itgrl clcultd i prt (): L 0 0 U 0

7 . (cotiud) d. Th dsird itrvl usig trms s spcifid is L U. Lt rprst th ctul sum of th giv sris: 0 First, comput th stimt of for th spcifid trms, which is th 0 th prtil sum : ,00,90 8, Nxt, us th xprssios for L U , L d U from prt (c) to vlut L d U : (Th ctul vlu of th sris is , which is i th itrvl obtid.) 600 0

8 6. 0 ic 0, th Root Tst idicts tht th sris covrgs. By th Root Tst, covrgs.

9 7. As, From th prcdig lysis, logicl choic for compriso sris is. Th it s of th rtio of th giv sris to th compriso sris must b computd: L ic 0 L, th Limit Compriso Tst idicts tht both sris ithr covrg or divrg. Th compriso sris is th divrgt Hrmoic ris. By th Limit Compriso Tst, divrgs.

10 ,96, ,7,6. To sum trms, th idx vlus 0 through must b usd. Th rmidr R is th costrid by R : , so th miimum umbr of trms rquird is 6. b. Comput th stimt of th vlu of th sris usig 6 trms, which is th th 6 prtil sum 6 :

11 8. (cotiud) c. Th ctul vlu of th sris is giv by 8,96, ,7,6 Th stimt obtid by ch prtil sum E r d th bsolut rror clcultd i squc for,,,, util E (ot tht ch prtil sum is obtid by ddig th corrspodig umbr of trms prviously computd i prt (b)): E E E E trms r rquird.

12 Extr Crdit 9. Bgi by computig svrl prtil sums to dvlop pttr: Th squc of prtil sums c b writt s follows: 6 7 8,,,,,,,, Nxt, lyz th squc of prtil sums: Numrtors: itgrs Domitors: multipls of Formul for th prtil sum:,,,,

13 0. l Bgi by usig th Altrtig ris Tst to dtrmi whthr or ot th giv sris covrgs: l l L'Hôpitl's Rul = 0 Th Altrtig ris Tsts idicts tht th giv sris covrgs. Nxt, bsolut covrgc is dtrmid by lyzig th sris of bsolut vlus of th trms of th giv sris for covrgc: For l l, ot tht l, so for sufficitly lrg vlus of, But, is p-sris with p, so it divrgs. l By th Compriso Tst, must lso divrg. Th giv sris covrgs, but th sris of bsolut vlus divrgs. l covrgs coditiolly. l.