JANE PROFESSOR WW Prob Lib1 Summer 2000

Transcription

1 JANE PROFESSOR WW Prob Lib Summer 000 Sample WeBWorK problems. WeBWorK assigmet Series6CompTests due /6/06 at :00 AM..( pt) Test each of the followig series for covergece by either the Compariso Test or the Limit Compariso Test. If either test ca be applied to the series, eter CONV if it coverges or DIV if it If either test ca be applied to the series, eter NA. (Note: this meas that eve if you kow a give series coverges by some other test, but the compariso tests caot be applied to it, the you must eter NA rather tha CONV.) cos ( pt) Test each of the followig series for covergece by either the Compariso Test or the Limit Compariso Test. If either test ca be applied to the series, eter CONV if it coverges or DIV if it If either test ca be applied to the series, eter NA. (Note: this meas that eve if you kow a give series coverges by some other test, but the compariso tests caot be applied to it, the you must eter NA rather tha CONV.) cos l 6 6 ( pt) Test each of the followig series for covergece by either the Compariso Test or the Limit Compariso Test. If at least oe test ca be applied to the series, eter CONV if it coverges or DIV if it If either test ca be applied to the series, eter NA. (Note: this meas that eve if you kow a give series coverges by some other test, but the compariso tests caot be applied to it, the you must eter NA rather tha CONV.) cos 4 ( pt) Test each of the followig series for covergece by either the Compariso Test or the Limit Compariso Test. If at least oe test ca be applied to the series, eter CONV if it coverges or DIV if it If either test ca be applied to the series, eter NA. (Note: this meas that eve if you kow a give series coverges by some other test, but the compariso tests caot be applied to it, the you must eter NA rather tha CONV.). l 6 7 cos ( pt) Each of the followig statemets is a attempt to show that a give series is coverget or ot usig the Compariso Test (NOT the Limit Compariso Test.) For each statemet, eter C (for correct ) if the argumet is valid, or eter I (for icorrect ) if ay part of the argumet is flawed. (Note: if the coclusio is true but the argumet that led to it was wrog, you must eter I.). For all,, ad the series coverges, so by the Compariso Test, the series coverges. For all, 3 9, ad the series coverges, so by the Compariso Test, the series 3 9 coverges. For all, l, ad the series co- verges, so by the Compariso Test, the series coverges. For all, l l, ad the series diverges, so by the Compariso Test, the series l For all, l, ad the series diverges, so by the Compariso Test, the series l

2 For all, 3, ad the series coverges, so by the Compariso Test, the series coverges. 3 ( pt) Each of the followig statemets is a attempt to show that a give series is coverget or ot usig the Compariso Test (NOT the Limit Compariso Test.) For each statemet, eter C (for correct ) if the argumet is valid, or eter I (for icorrect ) if ay part of the argumet is flawed. (Note: if the coclusio is true but the argumet that led to it was wrog, you must eter I.). For all,, ad the series coverges, so by the Compariso Test, the series coverges. For all, l, ad the series diverges, so by the Compariso Test, the series l For all, l, ad the series coverges, so by the Compariso Test, the series l coverges. For all, 3, ad the series coverges, so by the Compariso Test, the series 3 coverges. For all, arcta 3 π 3, ad the series π 3 coverges, so by the Compariso Test, the series arcta 3 coverges. For all, l, ad the series coverges, so by the Compariso Test, the series l coverges. 7.( pt) The three series A, B, ad C have terms A 9 B 3 C Use the Limit Compariso Test to compare the followig series to ay of the above series. For each of the series below, you must eter two letters. The first is the letter (A,B, or C) of the series above that it ca be legally compared to with the Limit Compariso Test. The secod is C if the give series coverges, or D if it So for istace, if you believe the series coverges ad ca be compared with series C above, you would eter CC; or if you believe it diverges ad ca be compared with series A, you would eter AD ( pt) The three series A, B, ad C have terms A 6 B 3 C Use the Limit Compariso Test to compare the followig series to ay of the above series. For each of the series below, you must eter two letters. The first is the letter (A,B, or C) of the series above that it ca be legally compared to with the Limit Compariso Test. The secod is C if the give series coverges, or D if it So for istace, if you believe the series coverges ad ca be compared with series C above, you would eter CC; or if you believe it diverges ad ca be compared with series A, you would eter AD ( pt) Select the FIRST correct reaso why the give series coverges. A. Coverget geometric series B. Coverget p series C. Compariso (or Limit Compariso) with a geometric or p series D. Coverges by alteratig series test. cos π l si 6 0.( pt) Select the FIRST correct reaso why the give series coverges. A. Coverget geometric series B. Coverget p series C. Compariso (or Limit Compariso) with a geometric or p series D. Caot apply ay test doe so far i class. 4

3 si 3 cos π l l e 7 cos π 3 7.( pt) Select the FIRST correct reaso why the give series coverges. A. Coverget geometric series B. Coverget p series C. Itegral test D. Compariso with a coverget p series E. Coverges by limit compariso test F. Coverges by alteratig series test. e π si ( pt) Select the FIRST correct reaso why the give series A. Diverges because the terms do t have limit zero B. Diverget geometric series C. Diverget p series D. Itegral test E. Compariso with a diverget p series F. Diverges by limit compariso test G. Diverges by alteratig series test. cos π l l 3 6 l 6 ( pt) A. Suppose that f(x) is a fuctio that is positive ad decreasig. Recall that by the itegral test: p f x dx p 0 f Recall that e Suppose that for each positive iteger k,! f k. Fid a upper boud B for k! f x dx B = B. A fuctio is give by h k x k e x dx 0. Its values may be foud i tables. Make the chage of variables y xl to express I x 4 x dx as a costat C times h 4 Fid C. 0 C = C. Let g x x 4 Fid the smallest umber M such that the fuctio g is decreasig for all x M C. M = D. Does 4 coverge or diverge? Aswer with oe letter, C or D. ( pt) For each sequece a fid a umber k such that k a has a fiite o-zero limit. (This is of use, because by the limit compariso test the series A. a B. a C. a a ad k both coverge or both diverge.) 7 3 D. a ( pt) For each sequece a fid a umber r such that a r has a fiite o-zero limit. (This is of use, because by the limit compariso test the series a ad A. a B. a C. a r both coverge or both diverge.) 8 3 7

4 D. a ( pt) The series 8 k r coverges whe 0 r ad diverges whe r This is true regardless of the value of the costat k. Whe r the series is a p-series. It coverges if k ad diverges otherwise. Each of the series below ca be compared to a series of the form k r For each series determie the best value of r ad decide whether the series coverges. A. 4 4 B. C. D. π 4 coverges or diverges (c or d)? 7 9 coverges or diverges (c or d)? 4 7 coverges or diverges (c or d)? coverges or diverges (c or d)? 7.( pt) For each of the series below select the letter from a to c that best applies ad the letter from d to k that best applies. A possible aswer is af, for example. A. The series is absolutely coverget. B. The series coverges, but ot absolutely. C. The series D. The alteratig series test shows the series coverges. E. The series is a p-series. F. The series is a geometric series. G. We ca decide whether this series coverges by compariso with a p series. H. We ca decide whether this series coverges by compariso with a geometric series. I. Partial sums of the series telescope. J. The terms of the series do ot have limit zero. K. Noe of the above reasos applies to the covergece or divergece of the series.. 3 si cos π π log 4!! 4 cos π π 8.( pt) For each of the series below select the letter from a to c that best applies ad the letter from d to k that best applies. A possible correct aswer is af, for example. A. The series is absolutely coverget. B. The series coverges, but ot absolutely. C. The series D. The alteratig series test shows the series coverges. E. The series is a p-series. F. The series is a geometric series. G. We ca decide whether this series coverges by compariso with a p series. H. We ca decide whether this series coverges by compariso with a geometric series. I. Partial sums of the series telescope. J. The terms of the series do ot have limit zero ! log log x dx ( pt) Select the FIRST correct reaso why the give series A. Diverges because the terms do t have limit zero B. Diverget geometric series C. Diverget p series D. Itegral test E. Compariso with a diverget p series F. Diverges by limit compariso test G. Caot apply ay test doe so far i class. 0 0 l!!

5 l Here is a short review of umerical series which you may fid helpful. REVIEW OF NUMERICAL SERIES SEQUENCES A sequece is a list of real umbers. It is called coverget if it has a limit. A icreasig sequece has a limit whe it has a upper boud. SERIES (Geometric series,ratioal umbers as decimals, harmoic series,divergece test) Give umbers formig a sequece a a! let us defie the th partial sum as sum of the first of them s a a! The SERIES is coverget if the SEQUENCE s s s 3 is.! I other words it coverges if the partial sums of the series approach a limit. A ecessary coditio for the covergece of this SERIES is that a s have limit 0. If this fails, the series The harmoic series +(/)+(/3)+... This illustrates that the terms a havig limit zero does ot guaratee the covergece of a series. A series with positive terms,i.e. a 0 for all, coverges exactly whe its partial sums have a upper boud. The geometric series r coverges exactly whe r INTEGRAL AND COMPARISON TESTS (Itegral test,p-series, compariso tests for covergece ad divergece, limit compariso test) Itegral test: Suppose f x is positive ad DECREASING for all large eough x. The the followig are equivalet: I. S. f x dx is fiite, i.e. coverges. f is fiite, i.e. coverges. This gives the p - test: coverges exactly whe p p Compariso test: Suppose there is a fixed umber K such that for all sufficietly large : 0 a Kb Covergece. If b coverges the so does a Divergece. If a diverges the so does (Positive series havig smaller terms are more likely to coverge.) Limit compariso test: SUPPOSE: a 0, b 0 ad a lim " b THEN R exists. Moreover, R is ot zero. a ad b both coverge or both diverge. OTHER CONVERGENCE TESTS FOR SERIES (Alteratig series test, absolute covergece, RATIO TEST) Alteratig series test: Suppose the sequece a a a 3 is decreasig ad has limit zero. The a coverges. This applies to ()-(/)+(/3)-(/4)+... Absolute Covergece Test: IF THEN Ratio test: SUPPOSE # a IF r IF r the the a coverges. a # has limit equal to r. a CONVERGES. a DIVERGES. a coverges, # # b. $ Prepared by the WeBWorK group, Dept. of Mathematics, Uiversity of Rochester, c UR