Series Solutions (BC only)

Transcription

1 Studet Study Sessio Solutios (BC oly) We have itetioally icluded more material tha ca be covered i most Studet Study Sessios to accout for groups that are able to aswer the questios at a faster rate. Use your ow judgmet, based o the group of studets, to determie the order ad selectio of questios to work i the sessio. Be sure to iclude a variety of types of questios (multiple choice, free respose, calculator, ad o-calculator) i the time allotted. Multiple Choice. D (97 BC9) I is a coverget p series, p II is the Harmoic series ad is kow to be diverget III is coverget by the Alteratig Test. C (98 BC) I is a coverget p series with p II is the Harmoic series which is diverget III is coverget by the Geometric Test with r. D (98 BC) The ceter is, so oly C,D, or E are possible. Check the edpoits: ( ) At : coverges by the Alteratig Test. At : which is the harmoic series ad kow to diverge.. A (99 BC6) I compare with p series, p so coverges 6 II coverges by the Geometric Test with r 7 III coverges sice it represets the alteratig harmoic series. C (99 BC7) This is a geometric series with Covergece for Coverget for. r. Copyright Natioal Math + Sciece Iitiative, Dallas, TX. All rights reserved. Visit us olie at

2 6. C (997 BC) Coverget by the Geometric Test sice a so the sum will be S. 8 r 8 Studet Study Sessio 7. E (997 BC) ( ). The edpoits of the iterval of covergece are whe ( ) ;,. Check edpoits: gives the alteratig harmoic series which coverges ad gives the Harmoic series which diverges. Therefore the iterval is 8. B (998 BC8) I is diverget sice the limit of the th term is ot zero. II is coverget sice it represets the alteratig harmoic series. III is diverget sice it represets the harmoic series. 9. A (998 BC) This is the itegral test applied to the series i (A). Thus the series i (A) coverges. Noe of the others must be true.. C ( BC) This is a Geometric i which a ad r so S. E ( BC) Simplify the terms with powers of to compare to p p for the compariso series to coverge by the p series test therefore p.. D ( BC) I coverges by the Geometric Test II diverges by the p series test sice p e III diverges by the th term test sice lim e. D (8 BC) By the Ratio Test, e! e e! e lim lim lim ( )! e ( )! e Copyright Natioal Math + Sciece Iitiative, Dallas, TX. All rights reserved. Visit us olie at

3 . D (8 BC) By the Ratio Test, e! e e! e lim lim lim ( )! e ( )! e Studet Study Sessio. D (8 BC6) By the Geometric Test, the series coverges whe This implies Therefore ( )( ) for ad oly.. 6. D (998 BC76) k must be odd i order for the first series to coverge by the Alteratig Test, ad k must be less tha i order for the secod series to coverge by the Geometric Test, therefore is the correct solutio. 7. B (998 BC8) Omit aswers D ad E sice they are cetered at ad the solutio should be cetered at. Check edpoits: coverges by the Alteratig Test ad diverges by the p series test therefore aswer B is correct. Copyright Natioal Math + Sciece Iitiative, Dallas, TX. All rights reserved. Visit us olie at

4 Studet Study Sessio Free Respose Solutios 8. BC6 (b) (c) lim lim At, the series is, which diverges. At, the series is, which diverges. Therefore, the iterval of covergece is. f lim lim... 9 f d d (d) The series represetig series. Therefore, f d. f d is a geometric : sets up ratio test computes limit : coclusio of ratio test : edpoit coclusio : aswer : atidifferetiatio of series : first three terms for defiite itegral series : geeral term : aswer Copyright Natioal Math + Sciece Iitiative, Dallas, TX. All rights reserved. Visit us olie at

5 9. BC6 lim lim for At, the series is which diverges sice this is the harmoic series. At, the series is which coverges by the Alteratig Test. Hece, the iterval of covergece is. f (b) : sets up ratio computes limit of ratio : idetifies iterior of iterval of covergece : aalysis/coclusio at edpoits : right edpoit : left edpoit if edpoits ot if multiple itervals : first terms : geeral term Studet Study Sessio (c) The series i b is a geometric series. f : substitutes ito b or ifiite series from epress series from b i closed form : aswer for studet s series OR f for. Therefore, 6 f Copyright Natioal Math + Sciece Iitiative, Dallas, TX. All rights reserved. Visit us olie at

6 . 6 BC6 lim The series coverges whe. Whe, the series is... This series does ot coverge, because the limit of the idividual terms is ot zero. Whe, the series is... This series does ot coverge, because the limit of the idividual terms is ot zero. Thus the iterval of covergece is. 9 f. g... ad g.!! 6! y f g (b) f... ad f ad g.! Thus, y. Sice y, y has a relative y ad miimum at. : sets up ratio : computes limit ratio : idetifies radius of covergece : cosiders both edpoits : aalysis/coclusio for both edpoits : y : y" : coclusio : reasoig Studet Study Sessio Copyright Natioal Math + Sciece Iitiative, Dallas, TX. All rights reserved. Visit us olie at

7 . 8B BC6 u u... u... u (b) No, the series does ot coverge whe because whe, the terms of the series do ot coverge to. (c) l t t dt 7... t t t t dt (d) l l... 7 Let A. Sice the series is a covergig alteratig series ad the absolute values of the idividual terms decrease to, A l. 6 6 Studet Study Sessio : two of the first four terms : remaiig terms : geeral term : aswer with reaso : two of the first four terms : remaiig terms : uses : value of A : justificatio Copyright Natioal Math + Sciece Iitiative, Dallas, TX. All rights reserved. Visit us olie at

8 . B BC6 : series : iterval of covergece Studet Study Sessio We must have so iterval of covergece is (b) : aswer (c) Some p such that because : correct p coverges by AST, but the : reaso why coverges p-series diverges for : reaso why diverges (d) Some p such that because the p-series diverges for ad the p-series coverges for : correct p : reaso why diverges : reaso why coverges Copyright Natioal Math + Sciece Iitiative, Dallas, TX. All rights reserved. Visit us olie at

9 . Form B BC6 Studet Study Sessio for : sets up ratio : limit evaluatio : radius of covergece : cosiders both edpoits : aalysis ad iterval of covergece Therefore the radius of covergece is. Whe the series is This is the harmoic series, which diverges. Whe the series is This is the alteratig harmoic series, which coverges. The iterval of covergece for the Maclauri series of f is (b) : series for : series for : series for : aalysis with geometric series is a geometric series that coverges to for Therefore for Copyright Natioal Math + Sciece Iitiative, Dallas, TX. All rights reserved. Visit us olie at