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1 Versio PREVIEW Homework Berg (5860 This pritout should have 9 questios. Multiplechoice questios may cotiue o the ext colum or page fid all choices before aswerig. CalCb0b poits Rewrite the fiite sum ( usig summatio otatio k = 5 k = 0 5 k = 0 8 k = 8 k = 8 k = k k ( k + k k + k k ( k + k k + k k ( k + correct k k ( k + The umerators form a sequece, 6, 0,,...,, while the deomiators form a sequece +, +, 5 +, 6 +,..., 8 +. Thus the geeral term i the series is of the form k k a k = ( k + where the sum rages from k = to k = 8. Cosequetly, the series becomes 8 k = i summatio otatio. k k ( k + CalCb0exam poits If the th partial sum of a is give by S = + +, what is a whe?. a =. a =. a =. a = 5. a = 6. a = By defiitio S = 5 ( + ( + 5 ( + ( + 5 ( + 9 ( + ( + correct 9 ( + ( + 9 ( + k Thus, for, a = S S Cosequetly, a = a = a + a a. = + + ( + ( +. 9 ( + ( +.
2 Versio PREVIEW Homework Berg (5860 CalCb5a poits Determie whether the series ( (cos π =0 is coverget or diverget, ad if coverget, fid its sum.. coverget with sum 9. diverget. coverget with sum 9 5 correct. coverget with sum coverget with sum 9 CalCbs poits Determie whether the ifiite series ( + ( + coverges or diverges, ad if coverges, fid its sum.. diverges correct. coverges with sum =. coverges with sum =. coverges with sum = 5. coverges with sum = 6. coverget with sum 9 5 Sice cos π = (, the give series ca be rewritte as a ifiite geometric series ( = a r i which =0 =0 a =, r =. But the series =0 ar is (i coverget with sum a whe r <, r ad (ii diverget whe r. Cosequetly, the give series is coverget with sum 9 5. By the Divergece Test, a ifiite series a diverges whe lim a 0. Now, for the give series, a = But the, ( + ( + = lim a = 0. Cosequetly, the Divergece Test says that the give series diverges. keywords: ifiite series, Divergece Test, ratioal fuctio.
3 Versio PREVIEW Homework Berg (5860 CalCbs poits Determie whether the ifiite series ta ( + 5 coverges or diverges, ad if it coverges, fid its sum.. coverges with sum = π. coverges with sum = π. diverges correct. coverges with sum = π 5. coverges with sum = π A ifiite series a diverges whe For the give series lim a 0. a = ta ( + 5 = ta ( + 5/ But lim ta ( + 5/ Cosequetly, the give series diverges. = ta ( = π. CalCbs 006 (part of 0.0 poits i Fid all values of x for which the series (x + =0 coverges. (. coverges oly o, (. coverges oly o,. coverges oly o (, 0. coverges oly o (0, ( 5. coverges oly o, correct ( 6. coverges oly o, The give series is a geometric series =0 with a = ad r = x +. Now such a geometric series i coverges whe r <, ad ii diverges whe r, so the give series coverges exactly whe < x + <. Thus it coverges oly o the iterval (,. 007 (part of 0.0 poits ii Fid the sum of the series (x + =0 for those values of x for which it coverges.. sum = x +. sum = x +. sum = x r
4 Versio PREVIEW Homework Berg (5860. sum = x 5. sum = x 6. sum = x + correct A geometric series r has sum = =0 a r whe it coverges. For a = ad r = x +, therefore, x +. CalCc0s poits Let g be a cotiuous, positive, decreasig fuctio o [,. Compare the values of the itegral ad the series B = 9 = A = 9 g(x dx g(, C = 8 = g(. I the figure a a a 5 a the bold lie is the graph of g o [, ad the areas of the rectagles the terms i the series. Clearly from this figure we see that while g( > g( > g(5 > g(6 > g(x dx, g(x dx, g(x dx, g(x dx, ad so o. O the other had, i the figure. B > C > A. B < A < C correct. B < C < A. A < B < C 5. C < A < B a a 5 a 6 a A > B > C the bold lie is agai the graph of g o [,,
5 Versio PREVIEW Homework Berg ( but ow while a = g( < a 5 = g(5 < a 6 = g(6 < a 7 = g(7 < ad so o. Cosequetly, B < A < C. CalCc0a poits g(x dx, g(x dx, g(x dx, g(x dx, Which of the followig series are coverget: A. B. + / 7. B ad C oly correct 8. oe of them By the Itegral test, if f(x is a positive, decreasig fuctio, the the ifiite series f( coverges if ad oly if the improper itegral f(x dx coverges. Thus for the three give series we have to use a appropriate choice of f. A. Use f(x = x +. The f(x dx is diverget (log itegral. B. Use f(x = x is coverget. /. The f(x dx C. Use f(x = x. The C is coverget. f(x dx. B oly. A ad B oly. C oly. A ad C oly 5. all of them 6. A oly keywords: coverget, Itegral test, CalCc0b poits Determie whether the series (l = is coverget or diverget.
6 Versio PREVIEW Homework Berg ( diverget correct. coverget By the Divergece Test, a series =N a will be diverget for each fixed choice of N if lim a 0 sice it is oly the behaviour of a as that s importat. Now, for the give series, N = ad a = (l. But by L Hospital s Rule applied twice, lim x x (lx = lim x = lim x x lx = lim x ( lx/x /x =. Cosequetly, by the Divergece Test, the give series is diverget. CalCc0c poits Determie whether the series k l(k k = is coverget or diverget.. series coverges. series diverges correct The fuctio f(x = x l(x is cotious, positive ad decreasig o [,. By the Itegral Test, therefore, the series k l(k k = coverges if ad oly if the improper itegral f(x dx = coverges, i.e., if ad oly if lim t t x l(x dx x l(x dx exists. To evaluate this last itegral, set u = l(x. The i which case, But t Cosequetly, du = x dx, l(t x l(x dx = l u du { } = l(l(t l(l. lim l(l(t =. t lim t t x l(x dx does ot exist. The Itegral Test thus esures that the give (A series diverges. CalCcexam poits Determie whether the followig series l(,
7 Versio PREVIEW Homework Berg ( (B + cos( + is coverget sice p = >. Thus, by the compariso test, series (B coverge or diverge. coverges.. A diverges, B coverges. both series diverge. both series coverge correct. A coverges, B diverges (A The fuctio f(x = l x x is cotious ad positive o [, ; i additio, sice ( lx f (x = < 0 o [,, f is also decreasig o this iterval. This suggests applyig the Itegral Test. Now, after Itegratio by Parts, we see that ad so t x [ f(x dx = l(x ] t x x, f(x dx = ( + l. The Itegral Test thus esures that series (A coverges. (B Note first that the iequalities 0 + cos( + + hold for all. O the other had, by the pseries test the series keywords: CalCds poits Determie whether the series k = coverges or diverges.. series is coverget k(k + (k + 5. series is diverget correct Note first that lim k k k(k + (k + 5 = > 0. Thus by the limit compariso test, the give series k(k + (k + 5 k = coverges if ad oly if the series k = coverges. But by the pseries test with p = (or use the compariso test applied to the harmoic series, this last series diverges. Cosequetly, the give k series is diverget. CalCd7s poits
8 Versio PREVIEW Homework Berg ( Determie whether the series coverges or diverges. ta 5 +. series is coverget correct. series is diverget We apply the Limit Compariso Test with For a = ta 5 +, b =. lim ( ta 5 + Thus the give series = lim ta = π. ta 5 + is coverget if ad oly if the series is coverget. But by the pseries test, this last series coverges because p = >. Cosequetly, the give series is coverget. CalCd9b poits Which of the followig ifiite series coverges?.. ( correct k = ( 7 7 ( + k lk + 7k We test the covergece of each of the five series. (i For the series k = k lk + 7k the limit compariso test ca be used, comparig it with k = k lk. Now, after divisio, ( k lk k lk + 7k = + 7 lk Sice + 7 > 0 lk as k, the limit compariso test applies. But by the Itegral test, the series k = does ot coverge. Thus k = does ot coverge. k lk k lk + 7k (ii If a series a coverges, the a 0.
9 Versio PREVIEW Homework Berg ( as. But for the series 7 ( + we see that a = 7 ( + = 7 ( + ( {( + = 7 = 7 + }. Now ( + e as. Thus Cosequetly, lim a = 7e 0. does ot coverge. (iii For the series 7 ( + 7 the compariso test ca be used sice 7 ( 7 while does ot coverge because of the Iegral test. Thus 7 does ot coverge. (iv Use of the Compariso test is suggested i dealig with ( + for after divisio Now the iequality + = 0 < + + holds for all, so the iequality ( + holds for all. But (. ( is a geometric series whose commo ratio is r =. Sice 0 < r <, this geometric series coverges. Hece by the Compariso test the series ( + coverges. (v The series ( 7 is a geometric series with r = 7 >. But the this geometric series does ot coverge. Cosequetly, of the five give ifiite series, oly coverges. ( + CalCds poits
10 Versio PREVIEW Homework Berg ( Determie whether the series ( si k k = coverges or diverges.. series is diverget correct. series is coverget We use the Limit Compariso test with ( a k = si, b k = k k. For the a k ad b k are series with positive terms ad ( si k lim k a k b k = lim k Thus the give series = lim θ 0 si θ θ k = k ( si k coverges if ad oly if the series k = k = > 0. coverges. But by the pseries test with p = (or because the harmoic series is diverget, this last series is diverget. Cosequetly, the give series is diverget. CalCda poits Which of the followig series (A 5 log + 6 (B (C diverge?. all of them. C oly 6 ( + ( 5 +. A ad B correct. A oly 5. A ad C (A We use the limit compariso test, comparig the give series with the series 5 log + 6 = log which by the Itegral test does ot coverge. Now ( log = 5 log log Sice log 5 > 0 as, the limit compariso test applies. Thus the give series i (A diverges. (B If a series a coverges, the a 0
11 Versio PREVIEW Homework Berg (5860 as. But for the give series, 6 ( a = ( + = 6 + ( {( + = 6 = 6 + }. Now ( + e as. Thus so the series diverges. lim a = 6e 0, (C After divisio, Now the iequality 6 ( = 5 +. Cosequetly, of the give ifiite series, diverge. oly A ad B CalCdb poits Which of the followig series coverge(s? k + (A k = (k lk (B + (C ( oe of A, B, or C. A ad B. A oly correct. C oly 5. B oly 0 < 5 6. A ad C A, B, ad C holds for all, so the iequality ( ( holds for all. But = ( 5 is a geometric series whose commo ratio is r = 5. Sice 0 < r <, this geometric series coverges. Hece by the compariso test the series ( 5 + coverges also. 8. B ad C (A After divisio, k + + (k lk + 5 = k k(lk + 5. k But + k(lk k k(lk + 5 > 0, k so k(lk ( k + (k lk + 5 > 0.
12 Versio PREVIEW Homework Berg (5860 O the other had, by the Itegral test the ifiite series k(lk k = coverges, hece by the limit compariso test, the give series i (A coverges also. (B If a ifiite series a coverges, the lim a = 0. But for the give series i (B, lim a = lim 7 + =. Cosequetly, the series i (B does ot coverge. (C After divisio, = 5, so the iequality ( holds for all. But the series ( 7 5 = ( 7 5 is a geometric series whose commo ratio r = 7 5. Now r >, so this geometric series does ot coverge. Hece by the compariso test the series i (C does ot coverge. Cosequetly, of the give ifiite series, coverges. oly A CalCd8a poits Fid all values of p for which the ifiite series ( 6 p coverges?. p <. p > correct. p < 5. p > 5 5. p > After divisio, so = 6 +, ( > 0 as. Thus by the Limit Compariso test, the ifiite series ( p will coverge if ad oly if the ifiite series p coverges. But by the Itegral test we kow that the series p coverges if ad oly if p >. Cosequetly, the give series will coverge if ad oly if p >, i.e., whe p >. 5 +
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