7) an = 7 n 7n. Solve the problem. Answer the question. n=1. Solve the problem. Answer the question. 16) an =

Transcription

1 Eam Name MULTIPLE CHOICE. Choose the oe alterative that best comletes the statemet or aswers the questio. ) Use series to estimate the itegral's value to withi a error of magitude less tha -.. l( + )d Determie if the sequece is decreasig or odecreasig ad if it is bouded or ubouded from above. ) a = + + Aswer the questio. Decreasig; bouded Decreasig; ubouded Nodecreasig; bouded Nodecreasig; ubouded ) The series below is the value of a Maclauri series of a fuctio f() at some oit. What fuctio ad what oit? e at = - at = + at = l + - at = Determie if the sequece is decreasig or odecreasig ad if it is bouded or ubouded from above. ( + )! ) a = ( + )! Decreasig; bouded Decreasig; ubouded Nodecreasig; bouded Nodecreasig; ubouded ) a = ()! Aswer the questio. Decreasig; bouded Decreasig; ubouded Nodecreasig; bouded Nodecreasig; ubouded ) The series below is the value of a Maclauri series of a fuctio f() at some oit. What fuctio ad what oit? (-)- l( + ) at = si at = ta- at = + at = ) ) ) ) ) ) Determie if the sequece is decreasig or odecreasig ad if it is bouded or ubouded from above. 7) a = 7 7 Decreasig; bouded Decreasig; ubouded Nodecreasig; bouded Nodecreasig; ubouded ) Use series to estimate the itegral's value to withi a error of magitude less tha -.. cos d Aswer the questio ) The series below is the value of a Maclauri series of a fuctio f() at some oit. What fuctio ad what oit? (-)- œ l( + ) at = cos at = ta- at = si at = ) Use series to estimate the itegral's value to withi a error of magitude less tha d.9... Aswer the questio. ) Estimate the error if si / is aroimated by / - 9/ i the itegral of! si / d. 7œ!! 7œ! œ! Determie if the sequece is decreasig or odecreasig ad if it is bouded or ubouded from above. ) a = ta- 7 7 Decreasig; bouded Decreasig; ubouded Nodecreasig; bouded Nodecreasig; ubouded ) a = l + + Decreasig; bouded Decreasig; ubouded Nodecreasig; bouded Nodecreasig; ubouded 7) ) 9) ) ) ) ) Use substitutio to fid the Taylor series at = of the give fuctio. ) l( + ) (-) (-)- ()! (-)- (-)- ) Use substitutio to fid the Taylor series at = of the give fuctio. ) e- () (-)() (-) (-)() () ) Aswer the questio. ) The series below is the value of a Maclauri series of a fuctio f() at some oit. What fuctio ad what oit? + + ta- at = + at = l + - at = si at = ) ) l (-)- + ) Determie if the sequece is decreasig or odecreasig ad if it is bouded or ubouded from above. si- ) a = ta- ) Determie if the sequece is decreasig or odecreasig ad if it is bouded or ubouded from above. ( + )! ) a = ( + )! Decreasig; bouded Decreasig; ubouded Nodecreasig; bouded Nodecreasig; ubouded ) Decreasig; bouded Decreasig; ubouded Nodecreasig; bouded Nodecreasig; ubouded 7) a = l Aswer the questio. Decreasig; bouded Decreasig; ubouded Nodecreasig; bouded Nodecreasig; ubouded ) For which of the followig is the corresodig Taylor series a fiite olyomial of degree? l() e- + - si 9) Use series to estimate the itegral's value to withi a error of magitude less tha -.. e- d ) ) 9) Aswer the questio. ) Estimate the error if e- is aroimated by - + i the itegral of! e- d. 7(!) (!) (!) 7(!) ) Which of the followig statemets are false? i. For a fuctio f(), the Taylor olyomial aroimatio ca always be imroved by icreasig the degree of the olyomial. ii. Of all olyomials of degree less tha or equal to, the Taylor olyomial of order gives the best aroimatio of f(). iii. The Taylor series at = a ca be obtaied by substitutig - a for i the corresodig Maclauri series. iii oly i, ii, ad iii i ad iii i ad ii Determie if the sequece is decreasig or odecreasig ad if it is bouded or ubouded from above. ) a = ta - ) ) ) Decreasig; bouded Decreasig; ubouded Nodecreasig; bouded Nodecreasig; ubouded

2 Determie if the series coverges or diverges; if the series coverges, fid its sum. ) (-)- 7 Coverges; Coverges; Coverges; Diverges l l 7) It ca be show that lim = for c >. Fid the smallest value of N such that < e for c c all > N if e =. ad c =.. ) Obtai the first ozero term of the Maclauri series for si - si - ta ) 7) ) Fid a formula for the th artial sum of the series ad use it to fid the series' sum if the series coverges. ) œ + œ + 9 7( + ) œ ( + ) ( + ) 7( + ) ; 7 ( + ) ( + ) ; 7 7 7( + )( + ) ; 7 ; 7 ( + )( + ) For what values of does the series coverge absolutely? ) + - < < - < < - < < - < < ) ) Fid the first four terms of the biomial series for the give fuctio. 9) - -/ ) Fid the Taylor series geerated by f at = a. ) f() =, a = 9 9 (l ) ( - 9) 9 (l ) ( - 9) 9 (l ) ( - 9) ( + )! 9 (l ) ( - 9) ( + )! ) Determie covergece or divergece of the alteratig series. ) (-) l + + Diverges Coverges ) Fid the limit of the sequece if it coverges; otherwise idicate divergece. ) a = Diverges ) Use the limit comariso test to determie if the series coverges or diverges. ) 7 + l coverges Diverges Fid the sum of the geometric series for those for which the series coverges. ) ( - ) ) ) Fid the Maclauri series for the give fuctio. 7) f() = e + ( + )! + + ( + )! Determie either absolute covergece, coditioal covergece or divergece for the series. ) (-) 7 - Coverges absolutely Diverges Coverges coditioally 7) ) 9) Let s k deote the kth artial sum of the alteratig harmoic series. Comute s 9 + s, s9 + s harmoic series? s 9 + s ) Use the fact that, ad s 9 + s. Which of these is closest to the eact sum (l ) of the alteratig cot = s 9 + s for < to fid the first four terms of the series for l(si ). s 9 + s l l ) ) ) A object is rollig with a drivig force that suddely ceases. The object the rolls meters i the first secod, ad i each subsequet iterval of time it rolls % of the distace it had rolled the secod before. This slowig is due to frictio. How far will the object evetually roll? It will roll a ifiite distace.. meters. meters. meters Use the root test to determie if the series coverges or diverges. ) / - Coverges Diverges Fid the sum of the geometric series for those for which the series coverges. 7) (-) ) ) 7) Use the itegral test to determie whether the series coverges. ) l coverges diverges ) ) To what value does the Fourier series of -, - < < f() =, < < coverge to whe =? ) For what values of does the series coverge absolutely? 9 ) < - < < < < - < < ) Let s k deote the kth artial sum of the alteratig harmoic series. If e() deotes the absolute value of the error i aroimatig l by s + s, comute floor deotes the iteger floor (or greatest iteger) fuctio., where floor() e() 9 7 ) ) Fid the smallest value of N that will make the iequality hold for all > N. 9). - < Fid the sum of the series as a fuctio of. ) ( + ) ) ) Fid the Taylor olyomial of lowest degree that will aroimate F() throughout the give iterval with a error of magitude less tha tha -. ) F() = e-t dt, [,.] ) Determie if the sequece is bouded. ) a = - bouded ot bouded Use series to evaluate the limit. si- - si ) lim ta- - ta ) )

3 Fid a series solutio for the iitial value roblem. ) y - y =, y() =, y () = 7 y = y = = = y = y = = = Determie covergece or divergece of the series. l () ) Aswer the questio. Diverges Coverges ) Which of the followig statemets is false? All of these are true. a If {a} ad {b} meet the coditios of the Limit Comariso test, the, if lim = ad b b coverges, the a coverges. The sequeces {a} ad {b} must be ositive for all to aly the Limit Comariso Test. The series a must have o egative terms i order for the Direct Comariso test to be alicable. Use the direct comariso test to determie if the series coverges or diverges. ) l + Diverges Coverges 7) Derive a series for l( + ) for > by first fidig the series for ad the itegratig. (Hit: + + = + /. ) (-)- (-)- l + l + - (-) (-) l + l + - ) ) ) ) 7) Fid the Taylor series geerated by f at = a. ) f() = , a = - ( + ) - ( + ) + 9( + ) - 9( + ) + ( + ) + ( + ) + 9( + ) - 9( + ) - 99 ( + ) + ( + ) + 9( + ) - 9( + ) + ( + ) - ( + ) + 9( + ) - 9( + ) - 99 Fid a series solutio for the iitial value roblem. 9) y + y =, y() = (-) y = y = (-)- y = y = Use the itegral test to determie whether the series coverges. ) + coverges diverges ) Usig the Maclauri series for l( + ), obtai a series for l( + ). (-) + + (-) + + (-)- (-) (-) + + (-) + + Determie either absolute covergece, coditioal covergece or divergece for the series. ) (-) () ( + )! coverges coditioally diverges coverges absolutely Use the itegral test to determie whether the series coverges. cos / ) diverges coverges Fid the Fourier series easio for the give fuctio. ) f() = si, - f() = si - f() = cos - f() = - f() = - cos si ) 9) ) ) ) ) ) 9 Fid the Maclauri series for the give fuctio. ) f() = e + + ( + )! ( + )! ) A recursio formula ad the iitial term(s) of a sequece are give. Write out the first five terms of the sequece. a 7) a =, a+ = 7) +,,,,,,,, Fid a formula for the th term of the sequece.,,,, 7,,,, 7 ),,,, (alteratig 's ad 's) ) a = + (-) + + (-)+ a = - (-) + (-) a = + (-) + (-) a = + (-) + - (-)+ Fid the sum of the series. 7) ) Use the direct comariso test to determie if the series coverges or diverges. 7) + Coverges Diverges 7) Use artial fractios to fid the sum of the series. 7) ( + )( + ) 7) Fid the sum of the series as a fuctio of. ( + ) ) - ( + ) + ( + ) - 9) For what value of r does the ifiite series + r + r + r + r + r + r + r7 + r +... coverge? r < r < 9 ( + ) + r < - ( + ) - r < ) 9) 7) If > ad q >, what ca be said about the covergece of? (l )q = Always diverges May coverge or diverge Always coverges Fid a formula for the th artial sum of the series ad use it to fid the series' sum if the series coverges. 7) œ + œ + œ ( + ) ( + ) ; ; ( + ) ( + ) ; + + ; 7) 7) 7) For aroimately what values of ca cos be relaced by - with a error of magitude o greater tha -? <.7 <. <. <.9 Write the first four elemets of the sequece. l ( + ) 7), l, l 7, l l, l, l 7, l, l 7, l, l l, l 7, l, l 7) 7) 77) A sequece of ratioal umbers r is defied by r =, ad if r = a b the r + = a + 7b a + b. Fid lim r. Hit: Comute the square of several terms of the sequece o a calculator. 7 7) A edulum is released ad swigs util it stos. If it asses through a arc of iches the first ass, ad if o each succesive ass it travels the distace of the recedig ass, how far will it travel before stoig?.7 i.. i..7 i. 7.7 i. 77) 7)

4 Use series to evaluate the limit. 79) lim e/ ) Use the direct comariso test to determie if the series coverges or diverges. e- 7) diverges coverges 7) ) To what value does the Fourier series of, - < < - f() =, - < <, < < coverge to whe = -? ) Fid the Maclauri series for the give fuctio. ) f() = l( + ) (-)- + + (-)- - + (-)- + (-) + + ) Fid the limit of the sequece if it coverges; otherwise idicate divergece. ) a = + e e Diverges Fid the first four terms of the biomial series for the give fuctio. ) ( + ) Determie covergece or divergece of the alteratig series. (-) ) + Diverges Coverges Fid the quadratic aroimatio of f at =. ) f() = si l( + ) Q() = - + Q() = + + Q() = - Q() = + Fid the sum of the geometric series for those for which the series coverges. - ) Determie if the sequece is bouded. ) - ) ) ) ) ) ) Fid the sum of the series. 9) + 7 Use the root test to determie if the series coverges or diverges. 9) l + 7 Coverges Diverges 9) If the series (-) ( - ) is itegrated twice term by term, for what value(s) of does the ew series coverge? < < < < Use the limit comariso test to determie if the series coverges or diverges. 9) + 7 (l ) Diverges Coverges Fid the first four terms of the biomial series for the give fuctio. 9) ( + )/ ) 9) 9) 9) 9) bouded ot bouded Fid the Maclauri series for the give fuctio. 9) f() = cos + (-) () ()! + + (-) () ()! + Fid the limit of the sequece if it coverges; otherwise idicate divergece. 9) a = (-) (-) () ()! (-) () ()! ± Diverges Use the itegral test to determie whether the series coverges. 7 9) e - coverges diverges Use the ratio test to determie if the series coverges or diverges. 97) Diverges Coverges Use series to evaluate the limit. e - - 9) lim Fid the values of for which the geometric series coverges. 99) ( + ) - < < - - < < - - < < - - < < Use the root test to determie if the series coverges or diverges. () ) ()! Coverges Diverges 9) 9) 9) 97) 9) 99) ) Fid the Fourier series easio for the give fuctio. -, - < < ) f() =, < < f() = f() = cos ( - ) - cos ( - ) - f() = f() = si ( - ) - si ( - ) - k ) If a is a coverget series of oegative terms, what ca be said about a, where k is a ositive iteger? Always diverges May coverge or diverge Always coverges Determie if the series coverges or diverges; if the series coverges, fid its sum. ( + ) si ) Coverges; Coverges; Determie if the series defied by the formula coverges or diverges. ) a = 7, a+ = a Coverges; Coverges Diverges Diverges ) For a alteratig series (-)+ u where lim u, what ca be said about the covergece or divergece of the series? The series always coverges. The series always diverges. The series may or may ot coverge. Use artial fractios to fid the sum of the series. ) ) ) ) ) ) )

5 7) If the series (-) ( - ) is itegrated term by term, for what value(s) of does the ew series coverge? < < < < Fid the sum of the series as a fuctio of. - ) Fid a formula for the th artial sum of the series ad use it to fid the series' sum if the series coverges. 9) (-) (-) (-)- + ; (-) + ; ; 7 - (-)- + ; 7) ) 9) Fid the limit of the sequece if it coverges; otherwise idicate divergece. ) a = œ e Diverges Fid the quadratic aroimatio of f at =. ) f() = - Q() = - Q() = + Q() = + Q() = - A recursio formula ad the iitial term(s) of a sequece are give. Write out the first five terms of the sequece. ) a =, a+ = a ) +, 7,,,, 7, 7,, 9, 7,, 9,, 7,,, ) If a ad b both coverge coditioally, what ca be said about ma a, b? The series always coverges. The series always diverges. The series may coverge or diverge. ) ) ) ) Use the fact that csc = (for < ) to fid the first four terms of the series for csc (for < ) ad thus for l ta. l l l l For what values of does the series coverge coditioally? (-) ( + ) ) = -, = - = - = - Noe Fid a formula for the th term of the sequece. ) -, -,,, (every other iteger startig with -) a = - a = - 7 a = - a = - ) ) ) Determie if the series coverges or diverges; if the series coverges, fid its sum. 7) ( ) Coverges; - Coverges; - Coverges; + Diverges ) If the series (-) ( - ) is itegrated term by term, for what value(s) of (if ay) does the ew series coverge ad for which the give series does ot coverge? = 7 = 9 = 7, = 9 oe For what values of does the series coverge absolutely? 9) () > < < - < < Noe 7) ) 9) 7 ) A sequece is defied by f() = floor( )œceilig( ). By lookig at several eamles o your calculator determie f( + ). + ( - ) ( + ) - ) For what values of does the series coverge coditioally? 9 7) = = 9 = - 9 Noe 7) Determie covergece or divergece of the series. ) = [l ()]/ Diverges Coverges Fid a formula for the th artial sum of the series ad use it to fid the series' sum if the series coverges. ) œ ; ; + ; series diverges 9 - ; series diverges - ) ) Determie covergece or divergece of the series. + 7 ) + + Diverges Coverges 9) For what value of r does the ifiite series + r + r + r + r + r + r +... coverge? r < r < r < r < ) 9) Write the first four elemets of the sequece. ) si (),, -,,,,,,,,,, - Use series to evaluate the limit. e - ) lim - Fid the Fourier series easio for the give fuctio., - < ) f() =, < f() = + cos ( - ) + ( - ) f() = - cos ( - ) - ( - ) f() = - si ( - ) - ( - ) f() = + si ( - ) + ( - ) (-) si (-) si (-) cos (-) cos ) Obtai the first ozero term of the Maclauri series for si- - ta-. - ) ) ) ) Determie if the series coverges or diverges; if the series coverges, fid its sum. ) Coverges; e coverges; e Fid the limit of the sequece if it coverges; otherwise idicate divergece. ) a = Fid the sum of the series as a fuctio of. + ) Coverges; Diverges Diverges A recursio formula ad the iitial term(s) of a sequece are give. Write out the first five terms of the sequece. ) a =, a+ = (-) + ) a,, -, -, -,,, -, -, -,, -,, -, -,, ) ) ) 9

6 Fid the sum of the geometric series for those for which the series coverges. ) si + si - si + si ) If a coverges coditioally, what ca be said about ma a, a? The series always coverges. The series always diverges. The series may coverge or diverge. Fid the sum of the geometric series for those for which the series coverges. ) (9 + ) si 9 ) ) ) ) Fid the sum of the series by eressig as a geometric series, differetiatig both 7- - sides of the resultig equatio with resect to, multilyig both sides by, differetiatig agai, ad relacig by Fid the Taylor olyomial of lowest degree that will aroimate F() throughout the give iterval with a error of magitude less tha tha -. si t ) F() = dt, [,.7] ) t ) A ball is droed from a height of m ad always rebouds of the height of the revious dro. ) ) Determie covergece or divergece of the alteratig series. 7) (-)+ + + Diverges Coverges Determie covergece or divergece of the series. ) si + + Coverges Diverges Fid the ifiite sum accurate to decimal laces. + 9) (-)+ ()! = Determie if the series defied by the formula coverges or diverges. + 9 si ) a =, a+ = 9 - cos a Coverges Diverges ) Derive the series for for > by first writig + + = + /. (-)+ (-) (-)+ (-) + + 7) ) 9) ) ) How far does it travel (u ad dow) before comig to rest? 9 m m 7 m m Fid a series solutio for the iitial value roblem. ) y - y =, y() = y = - + y = + + = = y = - + y = = = Fid the Taylor olyomial of lowest degree that will aroimate F() throughout the give iterval with a error of magitude less tha tha -. si t ) F() = dt, [, ] ) t Use the itegral test to determie whether the series coverges. 7) - diverges coverges ) 7) Fid the quadratic aroimatio of f at =. ) f() = l( + si 7) Q() = Q() = Q() = 7-9 Q() = Determie if the series coverges or diverges; if the series coverges, fid its sum. 9) e- e Coverges; Coverges; e - e- - e- Coverges; e- - Diverges ) 9) Determie if the series defied by the formula coverges or diverges. ) a = 9, a+ = + si a Coverges Diverges Use the direct comariso test to determie if the series coverges or diverges. si cos 7) 9 Diverges Coverges For what values of does the series coverge coditioally? ) + = = - = ± Noe ) 7) ) Fid the sum of the geometric series for those for which the series coverges. ) Fid the limit of the sequece if it coverges; otherwise idicate divergece. ) a = 9 9/ - ) ) Fid the smallest value of N that will make the iequality hold for all > N. 9). - < - 7 Determie if the series defied by the formula coverges or diverges. ) a =, a + = (a)+ Coverges Diverges 9) ) l 9 Diverges ) Obtai the first two terms of the Maclauri series for si(ta ) Fid the Taylor series geerated by f at = a. ) f() = - + -, a = - ( - ) + ( - ) + ( - ) + ( - ) - ( - ) + ( - ) - ( - ) + ( - ) + ( - ) + ( - ) - ( - ) + ( - ) - ) ) Fid the Taylor series geerated by f at = a. ) f() = e, a = e ( - ) e + ( - ) ( + )! Fid the quadratic aroimatio of f at =. ) f() = 9 - e + ( - ) e ( - ) ( + )! Q() = 9 Q() = Q() = - Q() = + ) ) Use artial fractios to fid the sum of the series. ) ta-(+) - ta-() - - ) Fid the iterval of covergece of the series. ) ( - ) < < 7-7 < < 7 < 7 < 7 ) Fid the quadratic aroimatio of f at =. ) f() = el Q() = Q() = + + Q() = + Q() = - )

7 Determie if the series coverges or diverges; if the series coverges, fid its sum. ) Coverges; - Coverges; - Coverges; + Diverges Fid a series solutio for the iitial value roblem. ) y - y =, y() = y = y = y = y = Determie if the series coverges or diverges; if the series coverges, fid its sum. cos ) 7 Coverges; 7 Coverges; 7 7) If > ad q >, what ca be said about the covergece of = (l ) q? Coverges; Diverges May coverge or diverge Always coverges Always diverges Determie either absolute covergece, coditioal covergece or divergece for the series. (-) ) 9/ + Coverges coditioally Coverges absolutely Diverges A recursio formula ad the iitial term(s) of a sequece are give. Write out the first five terms of the sequece. 9) a =, a+ = (-)a,, -, -, -,, -,, -, -,, -,, -, -,, 7) At a lat that ackages bottled srig water, the water is assed through a sequece of io-echage filters to reduce the sodium cotet rior to bottlig. Each filter removes % of the sodium reset i the water assig through it. Determie the umber of filters that must be used to reduce the sodium cocetratio from arts-er-millio to.9 arts-er-millio. 7 9 Fid the sum of the geometric series for those for which the series coverges. 7) ) ) ) 7) ) 9) 7) 7) Fid a formula for the th artial sum of the series ad use it to fid the series' sum if the series coverges. 7) œœ + œœ + œœ ( + )( + ) ) ( + ) ( + )( + ) ; ( + ) ( + )( + ) ; ( + ) ( + )( + ) ; ( + )( + ) ; ( + ) Use the limit comariso test to determie if the series coverges or diverges. 7 7) - l - Diverges Coverges 7) Give that the Fourier series of f() =, - < < is (-)- f() = si, what ca we say about the series obtaied by term-by-term differetiatio? The differetiated series diverges eve though its th term aroaches zero. The differetiated series coverges to g() =, - < <. The differetiated series coverges to somethig other tha the costat fuctio g() =, - < <. The differetiated series diverges sice its th term does ot aroach zero. By calculatig a aroriate umber of terms, determie if the series coverges or diverges. If it coverges, fid the limit L ad the smallest iteger N such that a - L <. for N; otherwise idicate divergece. 7) a = 7) L = l, N = L =, N = 7 L = l, N = diverges For what values of does the series coverge coditioally? ( + ) 7) = -7, = - = -7 = - Noe 77) It ca be show that for large values of. Fid the smallest value of N such that e - < - for all > N. e 9 7 7) 7) 7) 77) Fid the sum of the series. 7) = 7 7 Fid the ifiite sum accurate to decimal laces. + 79) (-)+ =, Fid the limit of the sequece if it coverges; otherwise idicate divergece. ) a = - - Diverges Determie whether the oicreasig sequece coverges or diverges. ) a = + œ Coverges Diverges 7) 79) ) ) ) Derive the series for for > by first writig + + = + /. (-)+ (-)+ (-) (-) + + Use the ratio test to determie if the series coverges or diverges. 7) Coverges Diverges Fid the Maclauri series for the give fuctio. ) f() = ta- (-) (-) (-) (-) ) 7) ) Fid the Fourier series easio for the give fuctio. ) f() =, - f() = - cos ( - ) ( - ) f() = - (-) cos () Chage the reeatig decimal to a fractio. ).... f() = - f() = - cos () (-)- cos ( - ) ( - ) By calculatig a aroriate umber of terms, determie if the series coverges or diverges. If it coverges, fid the limit L ad the smallest iteger N such that a - L <. for N; otherwise idicate divergece. ) a = / L =, N = L =, N = L =, N = 7 diverges ) Obtai the first ozero term of the Maclauri series for l( + ) - cos ) ) ) ) 9) Obtai the first ozero term of the Maclauri series for si - ta-. - 9) For aroimately what values of ca ta- be relaced by - + with a error of magitude o greater tha -? <.9 <.99 <.9 <. Fid the Maclauri series for the give fuctio. si 9) (-)+ + ( + )! (-) + ( + )! Determie if the sequece is bouded. 9) a = + + ot bouded bouded (-)+ + ( + )! (-) + ( + )! 9) 9) 9) 9) 7

8 Fid the Fourier series easio for the give fuctio., - < < - 9) f() = 9, - < <, < < ( + ) (-)- f() = 9 + cos - ( - ) (-)+ f() = 9 - cos - f() = 9 - f() = 9 + ( + ) (-)+ cos - ( - ) (-)- cos - 9) ) For a alteratig series (-)+ u where it is ot true that u u+ for sufficietly large, what ca be said about the covergece or divergece of the series? The series always coverges. The series always diverges. The series may or may ot coverge. ) Use series to evaluate the limit. cos - - 9) lim 9) Use the root test to determie if the series coverges or diverges. / - ) / - Coverges Diverges ) - - Determie covergece or divergece of the series. 9) ( + )/ Coverges Diverges 9) ) Fid the value of b for which - eb + eb - eb +... =. l l 9 l 7 l 9 ) Fid the iterval of covergece of the series. ( - ) 9) < < - < < < Use artial fractios to fid the sum of the series. 97) / - ( + )/ 9) Fid the sum of the ifiite series + r + r + r + r + r + r +... for those values of r for which it coverges. - r - r - r + r + r + r + r - r By calculatig a aroriate umber of terms, determie if the series coverges or diverges. If it coverges, fid the limit L ad the smallest iteger N such that a - L <. for N; otherwise idicate divergece. 99) a = cos L =, N = 7 L =, N = 9 L =, N = 9 diverges 9) 97) 9) 99) By calculatig a aroriate umber of terms, determie if the series coverges or diverges. If it coverges, fid the limit L ad the smallest iteger N such that a - L <. for N; otherwise idicate divergece. ) a = ta - e L = e, N = L =, N = L =, N = diverges e Fid the limit of the sequece if it coverges; otherwise idicate divergece. ) a = Fid the first four terms of the biomial series for the give fuctio. ) - / Diverges ) ) ) 9 Fid the Taylor olyomial of order geerated by f at a. ) f() =, a = + ( - ) + ( - ) + ( - ) + ( - ) + ( - ) + ( - ) + ( - ) + ( - ) + ( - ) + ( - ) + ( -) + ( - ) Fid the limit of the sequece if it coverges; otherwise idicate divergece. 7) a = ta - Diverges ) 7) Fid the Taylor olyomial of order geerated by f at a. ) f() = 9 -, a = Fid the values of for which the geometric series coverges. ) ( - ) < < - < < < < < < ) ) Use artial fractios to fid the sum of the series. ) l( + ) - l( + ) l l The series diverges. l ) ) To what value does the Fourier series of, - < < - f() =, - < <, < < coverge to whe =? ) Fid the first four terms of the biomial series for the give fuctio. 9) ( + 7)-/ ) Fid the sum of the series. [Hit: Write the series as ( - )( - ) ( - ) + + = ] = = = = 9e - e (e - ) e + Fid a formula for the th term of the sequece. ) 7,, 9,, (itegers begiig with 7) a = + a = + 7 a = + a = - 7 9) ) ) Determie either absolute covergece, coditioal covergece or divergece for the series. (-) ) 7 + Coverges absolutely Coverges coditioally Diverges Fid the limit of the sequece if it coverges; otherwise idicate divergece. ) a = l + l Diverges Fid a series solutio for the iitial value roblem. 7) y - y + 9y =, y() =, y () = + y = y = ( + )! y = y = + + ) ) 7)

9 ) Usig the Maclauri series for l( + ), obtai a series for l (-) + (-) Aswer the questio. 9) Which of the followig is ot a coditio for alyig the itegral test to the sequece {a}, where a = f()? All of these are coditios for alyig the itegral test. f() is cotiuous for N f() is decreasig for N f() is everywhere ositive Fid the iterval of covergece of the series. ( - ) ) l ( + ) < 7 < 7 - < < < < 7 ) A sequece is defied by f() = floor( )œceilig( ). By lookig at several eamles o your calculator determie f( + ). + ( + ) - ( - ) Fid the Taylor olyomial of order geerated by f at a. ) f() = +, a = ) 9) ) ) ) Fid the first four terms of the biomial series for the give fuctio. ) By calculatig a aroriate umber of terms, determie if the series coverges or diverges. If it coverges, fid the limit L ad the smallest iteger N such that a - L <. for N; otherwise idicate divergece. ) a = + L =, N = 9 L = l, N = 9 L =, N = diverges Use series to evaluate the limit. si 7 - si ) lim ) Fid the value of b for which + eb + eb + eb +... =. l l Fid the Taylor series geerated by f at = a. ) f() = 9 -, a = 7 l l ( - 7) ( - 7)+ ( - 7) + + Determie if the sequece is bouded. 9) bouded ot bouded ( - 7)+ ) ) ) 7) ) 9) ) It ca be show that lim c = for c >. Fid the smallest value of N such that c all > N if e =. ad c =.. < e for ) Fid the sum of the series. ) ) ) If cos is relaced by - ad <.7, what estimate ca be made of the error? E <.9 E <.7 E <. E <.77 ) Fid the limit of the sequece if it coverges; otherwise idicate divergece. ) a = + (-) + (-)(+) Diverges Fid the values of for which the geometric series coverges. ) (-) () ) ) Fid a formula for the th artial sum of the series ad use it to fid the series' sum if the series coverges. ) (-)- 7 œ (-) ; series diverges 7 - (-) ; (-) - ; 7 - (-) ; series diverges - Fid the limit of the sequece if it coverges; otherwise idicate divergece. ) < < < < ) a = (-) - ) Fid the first four terms of the biomial series for the give fuctio. ) ( - 7)/ Use the itegral test to determie whether the series coverges. ) diverges coverges Fid the Maclauri series for the give fuctio. ) e- (-) (-) ) ) ) Diverges Fid the first four terms of the biomial series for the give fuctio. ) ( + )/ Use the root test to determie if the series coverges or diverges. 7 ) l + + Diverges Coverges Fid the sum of the series. 7 ) 7 7 ) ) ) Estimate the magitude of the error ivolved i usig the sum of the first four terms to aroimate the sum of the etire series. 7) (-)+ 7). - Use the direct comariso test to determie if the series coverges or diverges. ) - + coverges diverges Fid the smallest value of N that will make the iequality hold for all > N. 9) < ) 9) Determie covergece or divergece of the alteratig series. ) (-)+ ( + ) / / + Coverges Diverges Fid the limit of the sequece if it coverges; otherwise idicate divergece. ) a = 7 + (.9) Diverges Use the root test to determie if the series coverges or diverges. 7) / + Diverges Coverges ) ) 7)

10 Fid the sum of the geometric series for those for which the series coverges. ) (-) () ) Fid the sum of the series. ) (-) ) Determie if the series defied by the formula coverges or diverges. 9) Let f() = What is f() aroimately equal to as gets large? Hit: Comute various eamles o your calculator. Use the ratio test to determie if the series coverges or diverges. ) Diverges Coverges 9) ) 7) a = 7, a + = a Coverges Diverges ) A child o a swig swees out a distace of ft o the first ass. If she is allowed to cotiue swigig util she stos, ad if o each ass she swees out a distace of the revious ass, how far does the child travel? ft ft ft ft 7) ) Determie either absolute covergece, coditioal covergece or divergece for the series. ) (cos ) diverges coverges absolutely coverges coditioally ) Fid a formula for the th term of the sequece. 9),,,,,,, (alteratig 's ad 's i airs) (+) a = + (-) a = + (-)(+) 9) Determie covergece or divergece of the series. + 7 ) Coverges Diverges ) a = - (-)(-) a = + (-)(-) Fid the first four terms of the biomial series for the give fuctio. ) + - ) Use the limit comariso test to determie if the series coverges or diverges. (l ) ) ( + ) Coverges Diverges Fid the limit of the sequece if it coverges; otherwise idicate divergece. ) a = (l ) e l Diverges A recursio formula ad the iitial term(s) of a sequece are give. Write out the first five terms of the sequece. ) a =, a+ = a, 7,, 9,,,, 9, 777,,,, 9, 7,, 9, ) ) ) Fid the Taylor olyomial of lowest degree that will aroimate F() throughout the give iterval with a error of magitude less tha tha -. ta- t ) F() = dt, [,.] ) t Fid the smallest value of N that will make the iequality hold for all > N. ) - < ) 7 For what values of does the series coverge coditioally? ) ( + ) = - = - = -7 Noe Fid the Fourier series easio for the give fuctio., - < < ) f() =, < < f() = - f() = + cos ( - ) - cos ( - ) - Fid a formula for the th term of the sequece. f() = - f() = + ), -, 9, -, (recirocals of squares with alteratig sigs) a = (-) a = (-) a = (-) + si ( - ) - si ( - ) - a = (-) + ) ) ) Chage the reeatig decimal to a fractio. 7) Determie covergece or divergece of the series. / 7) / Diverges Coverges 99 By calculatig a aroriate umber of terms, determie if the series coverges or diverges. If it coverges, fid the limit L ad the smallest iteger N such that a - L <. for N; otherwise idicate divergece. 7) a = 7) L = l, N = L = l, N = L =, N = diverges Determie whether the oicreasig sequece coverges or diverges. 7) a =, a+ = a - 7 Diverges Coverges 7) 7) 7) Fid the Taylor series geerated by f at = a. ) f() =, a = 9 (-) ( - 9) 9+ (-) ( + )( - 9) 9+ For what values of does the series coverge absolutely? 7) + 7 (-) ( - 9) 9+ (-) ( + )( - 9) < - < < - < Fid the limit of the sequece if it coverges; otherwise idicate divergece. ) a = + e Diverges Fid the sum of the series. 9) - (-) 7 7 ) 7) ) 9) Fid the quadratic aroimatio of f at =. 7) f() = ta- 7 Q() = + 7 Q() = - 7 Q() + 7 Q() = 7 For what values of does the series coverge absolutely? 7) ( - ) < < < < 7) If a coverges, what ca be said about a a+? The series always coverges. The series always diverges. The series may coverge or diverge. 77) Use the fact that ta- = (-)- - ( - ) for < to fid the series for cot-. - (-)- - ( - ) (-) - ( - ) - (-)- - ( - ) (-) - ( - ) 7) 7) 7) 77) 9

11 Fid the ifiite sum accurate to decimal laces. + (-)+ 7) =.... 7) Use the limit comariso test to determie if the series coverges or diverges. (l ) ) ( + ) Diverges Coverges ) Fid the quadratic aroimatio of f at =. 79) f() = l(cos 9) Q() = - Q() = Q() = + Q() = - For what values of does the series coverge coditioally? ) () = = = - Noe 79) ) Use the itegral test to determie whether the series coverges. 7 ) diverges coverges 7) If ta- is relaced by - + ad <., what estimate ca be made of the error? E <. E <. E <. E <. ) 7) Fid the Maclauri series for the give fuctio. ) cos (-) (-) - ()! (-) - ()! (-) ()! (-) ()! By calculatig a aroriate umber of terms, determie if the series coverges or diverges. If it coverges, fid the limit L ad the smallest iteger N such that a - L <. for N; otherwise idicate divergece. ) a = ta L =, N = L =, N = L =, N = diverges ) ) Fid a series solutio for the iitial value roblem. ) y + y =, y() =, y () = y = + + (-) y = - ()! + y = - + (-)- y = + ()! + Use the direct comariso test to determie if the series coverges or diverges. 9) (l ) Diverges Coverges (-) ()! (-)- ()! ) 9) Use the direct comariso test to determie if the series coverges or diverges. + 9 cos ) Diverges Coverges ) 9) Fid the sum of the series (-)- by eressig as a geometric series, - + differetiatig both sides of the resultig equatio with resect to, ad relacig by. 9) Fid a formula for the th term of the sequece. ),,,,,,, ( ad 's reeated) (+)(+) a = + (-) a = + (-) (+) a = + (-) a = + (-) (+)(+) (-) ) Fid the Taylor olyomial of order geerated by f at a. 9) f() = +, a = ( - ) ( - ) + ( + ) - ( + ) ( - ) 9 + ( + ) - ( + ) 9 - ( - ) 9 9) 9) f() =, a = + ( - ) + ( - ) + ( - ) + ( - ) + ( - ) + ( - ) + ( - ) + ( - ) + 7( - ) + ( - ) + ( - ) Use the itegral test to determie whether the series coverges. 9) e - Aswer the questio. diverges coverges 9) Which of the followig sequeces do ot meet the coditios of the Itegral Test? i). a = (si +) ii). a = + iii). a = i ad iii ii ad iii i oly, ii, ad iii 9) If a is a coverget series of oegative terms, what ca be said about a? Always coverges Always diverges May coverge or diverge Fid the Taylor olyomial of lowest degree that will aroimate F() throughout the give iterval with a error of magitude less tha tha -. 9) F() = (ta- t) dt, [,.] 9) Use artial fractios to fid the sum of the series. 7 97) ( + ) ) 9) 9) 9) 97) Fid the Fourier series easio for the give fuctio. 9) f() = cos, - < < f() = cos - f() = si - f() = - si + f() = cos - = = = = (-) si - (-) si - (-) si - (-) si - 99) If a coverges coditioally, what ca be said about mi a, a? The series always coverges. The series always diverges. The series may coverge or diverge. Determie covergece or divergece of the series. ) e- Diverges Coverges By calculatig a aroriate umber of terms, determie if the series coverges or diverges. If it coverges, fid the limit L ad the smallest iteger N such that a - L <. for N; otherwise idicate divergece. ) a = cos L =, N = L =, N = 7 L =, N = diverges Use artial fractios to fid the sum of the series. ) ) 99) ) ) )

12 ) Use the fact that si- = + œ œ œ... œ ( - ) + œ œ œ... œ ()( + ) for < to fid the series for cos œ œ œ... œ ( - ) + œ œ œ... œ ()( + ) - + (-) œ œ œ œ... œ ( - ) + œ œ œ... œ ()( + ) - + (-)+ œ œ œ œ... œ ( - ) + œ œ œ... œ ()( + ) - œ œ œ... œ ( - ) + œ œ œ... œ ()( + ) ) 9) If a coverges, what ca be said about a k, where k is a iteger greater tha? The series always coverges. The series always diverges. The series may coverge or diverge. ) Derive a series for l( + ) for > by first fidig the series for ad the itegratig. + (Hit: + = + /. ) (-)- (-)+ l + l + (-) (-)+ l + l + 9) ) Fid the limit of the sequece if it coverges; otherwise idicate divergece. ) a = + (-) 9 Diverges ) For what values of does the series coverge absolutely? (-)+ ( + ) ) 7 < - < < - - < - - < < - ) For what values of does the series coverge coditioally? (-)+ ( + ) ) = -, = = = - Noe A recursio formula ad the iitial term(s) of a sequece are give. Write out the first five terms of the sequece. ) a =, a =, a+ = a+ - a 7) Let,,,,,,, -, -, -,, -,, -,, -, ad f() = g() =, - < <, < <, - < <, < <. To what value does the Fourier series of f() + g() coverge to whe =? Use the direct comariso test to determie if the series coverges or diverges. ) + 9 Coverges Diverges ) ) 7) ) ) Mari dros a ball from a height of meters ad otices that o each bouce the ball returs to about / of its revious height. About how far will ball travel before it comes to rest?. meters meters meters meters Fid the values of for which the geometric series coverges. ) si - < < is ot a odd multile of / is ot a multile of diverges for all Use the root test to determie if the series coverges or diverges. ) - 9 Coverges Diverges Use artial fractios to fid the sum of the series. 7 ) ( - )( + ) ) ) ) ) Fid the smallest value of N that will make the iequality hold for all > N. ) < - ) ) The olyomial + + is used to aroimate f() = ( + ) o the iterval -... Use the Remaider Estimatio Theorem to estimate the maimum absolute error ) By calculatig a aroriate umber of terms, determie if the series coverges or diverges. If it coverges, fid the limit L ad the smallest iteger N such that a - L <. for N; otherwise idicate divergece. 7) a = ( +./) L., N = L., N = L., N = diverges Fid the first four terms of the biomial series for the give fuctio. ) ( - ) -/ Determie either absolute covergece, coditioal covergece or divergece for the series. 9) (-)- diverges coverges absolutely coverges coditioally ) For aroimately what values of ca cos be relaced by - + with a error of magitude o greater tha -? <.9 <. <.9 <.9 A recursio formula ad the iitial term(s) of a sequece are give. Write out the first five terms of the sequece. ) a =, a+ = a +,,,,,,, 9,, 7,,, 9,, 7, 9,, 7, Fid the values of for which the geometric series coverges. - ) - < < < < - < < - < < 7) ) 9) ) ) ) Fid a formula for the th term of the sequece. ), -,,,, -,, (, -,, reeated) a = cos() a = si() a = cos ),,,, (alteratig 's ad 's) a = si a = - (-) a = + (-) a = + (-)- a = + (-)+ Use the ratio test to determie if the series coverges or diverges. () 7) ()! Coverges Diverges By calculatig a aroriate umber of terms, determie if the series coverges or diverges. If it coverges, fid the limit L ad the smallest iteger N such that a - L <. for N; otherwise idicate divergece. ) a = cos ) Aswer the questio. L =, N = L =, N = L =, N = diverges 9) Which of the followig statemets is false? If a ad f() satisfy the requiremets of the Itegral Test, ad if f()d coverges, the a = f()d coverges if > ad diverges if. The itegral test does ot aly to diverget sequeces. coverges if >. = (l ) ) ) 7) 9) Fid the Taylor series geerated by f at = a. ) f() = e, a = 9 e9 ( - 9) ( + )! e9 ( - 9)+ e9 ( - 9)+ ( + )! e9 ( - 9) ) ) If > ad q >, what ca be said about the covergece of (l ) = (l l ) q? Always coverges May coverge or diverge Always diverges ) 7

13 Fid the ifiite sum accurate to decimal laces. + ) (-) ( + ) = Determie covergece or divergece of the alteratig series. (-) ) / Diverges Coverges Fid the values of for which the geometric series coverges. (-) ) ( + si ) - < < is ot a multile of diverges for all is ot a multile of Fid the limit of the sequece if it coverges; otherwise idicate divergece. ) a = l Diverges Fid the values of for which the geometric series coverges. ) - < < < Fid the Maclauri series for the give fuctio. ) f() = ( - ) (+) + + (+) + + < 7) A comay adots a advertisig camaig to weekly add to its customer base. It assumes that as a average fifty ercet of its ew customers, those added the revious week, will brig i oe fried, but those who have bee customers loger will ot be very effective as recruiters ad ca be discouted. A media camaig brigs i, customers iitially. What is the eected total umber of customer with whom the comay ca eect to have dealigs?,, The sum diverges to ifiity., ) ) ) ) ) ) 7) Fid the smallest value of N that will make the iequality hold for all > N. ) < - 7 Determie covergece or divergece of the alteratig series. 9) (-) + + Coverges Diverges Fid a series solutio for the iitial value roblem. ) y + 7y =, y() =, y () = -7 (-)- 7 y = + y = + (-)- 7 y = + y = - + = Fid the ifiite sum accurate to decimal laces. + ) (-)+ = (-) 7 (-) ) A comay's aual reveue for the eriod sice ca be modeled by the fuctio R =.(.), where R is i millios of dollars ad = corresods to. Assumig the model accurately redicts future reveue, fid the year i which the reveue first eceeds $. millio. 9 Use the root test to determie if the series coverges or diverges. () ) Diverges Coverges Fid the Taylor olyomial of lowest degree that will aroimate F() throughout the give iterval with a error of magitude less tha tha -. l( + t) ) F() = dt, [,.] ) t ) 9) ) ) ) ) 9 Fid a formula for the th artial sum of the series ad use it to fid the series' sum if the series coverges. 7 ) œ + 7 œ + 7 œ ( + )( + ) ; 7 7 ( + ) ; 7 7 ( + ) ; ; 7 ) ) If a is a coverget series of oegative terms, what ca be said about k a, where k is a ositive iteger? Always coverges Always diverges May coverge or diverge ) Fid the Taylor olyomial of order geerated by f at a. ) f() = l( + ), a = l + - l ( - ) + ( - ) 7 + ( - ) - ( - ) 7 Use the itegral test to determie whether the series coverges. 7) cos-(/) l l diverges coverges Fid the Taylor olyomial of order geerated by f at a. ) f() = l, a = l - - l ( - ) + ( - ) - ( - ) 9 + ( - ) 9 l - - l ( - ) + ( - ) ( - ) - ( - ) ( - ) - ( - ) - ( - ) + ( - ) ) 7) ) Use series to evaluate the limit. + l( + ) - cos ) lim 9 Fid the sum of the series. ) - - Determie if the series coverges or diverges; if the series coverges, fid its sum. ) + - Coverges; Coverges; e- - - Coverges; - + Diverges - ) ) ) Determie either absolute covergece, coditioal covergece or divergece for the series. 9) (-)( + - ) coverges absolutely coverges coditioally diverges For what values of does the series coverge absolutely? ) (-) ( + ) - < - - < < - Determie if the series defied by the formula coverges or diverges. ) a =, a+ = + a Coverges Diverges Chage the reeatig decimal to a fractio. ) < < - - < ) ) ) ) Fid the Taylor olyomial of order geerated by f at a. 7) f() = 7 -, a = ( - ) + ( + ) - ( + ) 9 - ( - ) 9 Fid the quadratic aroimatio of f at =. ) f() = esi ( - ) - + Q() = - + Q() = - Q() = + + Q() = + Use the limit comariso test to determie if the series coverges or diverges. + 7 si 9) 9/ + cos Coverges Diverges - ( - ) 9 + ( + ) - ( + ) 9 7) ) 9)

14 ) Let s k deote the kth artial sum of the alteratig harmoic series. Comute s9, s, ad ) Fid the limit of the sequece if it coverges; otherwise idicate divergece. 7) a = l(9 - ) - l( + ) 7) s9 + s. Which of these is closest to the eact sum (l ) of the alteratig harmoic series? l 9 l 9 l Diverges s9 s s 9 + s ) A child o a swig iitially swigs through a arc legth of meters. The child stos ushig ad sits atietly waitig for the swig to sto movig. If frictio slows the swig so the legth of each arc is % of the legth of the revious arc, how far will the child have traveled before the swig stos? meters meters The child will travel a ifiite distace. meters ) ) Fid the sum of the series (-)- by eressig as a geometric series, - + differetiatig both sides of the resultig equatio with resect to, multilyig both sides by, differetiatig agai, ad relacig by. 9 ) Determie if the series coverges or diverges; if the series coverges, fid its sum. ) l 7 Coverges; Coverges; l 7 Fid the Maclauri series for the give fuctio. ) - Coverges; l 7 Diverges Use series to evaluate the limit. ) lim si - ta ) ) ) Fid the Taylor series geerated by f at = a. 9) f() = + -, a = - ( + ) + ( + ) - ( - ) + ( - ) - ( - ) + ( - ) - ( + ) + ( + ) - Estimate the magitude of the error ivolved i usig the sum of the first four terms to aroimate the sum of the etire series. (-)+(.)+ 7) 7) Use the direct comariso test to determie if the series coverges or diverges. 7) + 7 Diverges Coverges 9) 7) For what values of does the series coverge coditioally? ) (-) (9 + ) = - = - 9 = - 9 Noe ) Chage the reeatig decimal to a fractio. 7) ) If si is relaced by - ad <.7, what estimate ca be made of the error? 7) 7) For what values of does the series coverge absolutely? ) = (l ) = - < < - < ) E <.79 E <. E <. E <. 7) A sequece of ratioal umbers r is defied by r =, ad if r = a b the r + = a + b a - b. Fid r. 7) 9 Fid the iterval of covergece of the series. ( - 9) 7) < < 97 < < < 9 7) If cos is relaced by - + ad <., what estimate ca be made of the error? E <. E <.7 E <. E <. 77) Fid the sum of the series. [Hit: Write the series as ( - ) + = + +.] = = = (e - ) e e - e + Determie covergece or divergece of the alteratig series. (-) 7) + Diverges Coverges 7) 7) 77) 7) Determie if the sequece is bouded. ) e bouded ot bouded For what values of does the series coverge coditioally? ) + = - = = ± Noe Fid the smallest value of N that will make the iequality hold for all > N. ) - < Use the limit comariso test to determie if the series coverges or diverges. ) l l = Diverges Coverges ) A ball is droed from a height of meters. If each bouce brigs it to 9% of its revious height, how far will the ball travel before it stos? meters 9 meters meters meters ) ) ) ) ) For what values of does the series coverge absolutely? (-) ( - ) 79) 9 < 9 < 9 9 < < 79) Chage the reeatig decimal to a fractio. ) ) Fid a series solutio for the iitial value roblem. ) y + y =, y() = -, y () = y = y = y = y = = + = + = + = (-) (-)- (-)- (-) ) For what values of does the series coverge coditioally? 7) = (l )7 = = - = ± Noe Determie if the series coverges or diverges; if the series coverges, fid its sum. + ) 9- Coverges; Coverges; Coverges; Diverges 9) Use a grahical method to determie the aroimate iterval for which the secod order Taylor olyomial for l ( + ) at = aroimates l ( + ) with a absolute error of o more tha ) ) 9)

15 Fid the sum of the series as a fuctio of. + 9) + - Fid the values of for which the geometric series coverges. 9) ( + ) - < < - < < - - < < - + < < 9) 9) Fid the Maclauri series for the give fuctio. 97) si (-)+ + + (-)+ + + ( + )! Fid a formula for the th term of the sequece. 9) -9, -, -7, -, - (itegers begiig with -9) (-) + + ( + )! (-) + + a = - 9 a = + 9 a = - a = - 97) 9) Fid the iterval of covergece of the series. ( - ) 9) ()! - < < - 7 Use the ratio test to determie if the series coverges or diverges. ()! 9) Diverges Coverges 9) 9) Fid the Taylor olyomial of order geerated by f at a. 99) f() = e-, a = - + -, ) f() = + +, a = ( - ) + ( - ) + ( - ) + 9( - ) + ( - ) + 9( - ) + 9( - ) + ( - ) + 9( - ) + ( - ) 99) ) Fid the Maclauri series for the give fuctio. 9) e7 7 (-) 7 9) f() = + 7 (-) 7 7 (-)+ 7 + (-) 7 + (-)+ 7 + (-) 7 + Use the root test to determie if the series coverges or diverges. l 9) 7 + Diverges Coverges 9) 9) 9) Fid the limit of the sequece if it coverges; otherwise idicate divergece. ) a = œ Diverges ) You la to estimate e by evaluatig the Maclauri series for f() = e at =. How may terms of the series would you have to add to be sure the estimate is good to decimal laces? 9 ) If a is a coverget series of oegative terms, what ca be said about a k, where k is a ositive iteger? May coverge or diverge Always coverges Always diverges Fid the sum of the geometric series for those for which the series coverges. (-) ) (9 + si ) 9 + si 9 + si + si ( + si ) ( - si ) 9( - si ) + si 9( + si ) ) ) ) ) 7 ) A comay makes a very durable roduct. It sells, i the first year, but will have dimishig sales due to the roduct's durability, so that each year it ca eect to sell oly sevety-five ercet of the quatity it will have sold the year before. How may of the roduct ca the comay eect to evetually sell?,,,,7 ) Estimate the magitude of the error ivolved i usig the sum of the first four terms to aroimate the sum of the etire series. (-)+(.) ) ) Use the root test to determie if the series coverges or diverges. ) 9 Coverges Diverges ) Use the limit comariso test to determie if the series coverges or diverges. ) / Diverges Coverges ) Aswer the questio. a 7) Suose that a > ad b > for all N (N a iteger). If lim =, what ca you coclude b about the covergece of a? a diverges if b diverges The covergece of a caot be determied. a diverges if b diverges, ad a coverges if b coverges a coverges if b coverges 7) Determie if the sequece is bouded. ) a = ot bouded bouded ) For aroimately what values of ca si be relaced by - + with a error of magitude o greater tha -? <.7 <.797 <.7 <.9 ) ) ) If > ad q >, what ca be said about the covergece of = (l ) q? Always coverges Always diverges May coverge or diverge Use the itegral test to determie whether the series coverges. 9) coverges diverges ) 9) Determie either absolute covergece, coditioal covergece or divergece for the series. ) (-) Coverges absolutely Diverges Coverges coditioally 7) Usig the Maclauri series for ta-, obtai a series for ta -. (-) (-) (-) (-) + ) 7) Determie covergece or divergece of the series. ) e- Coverges Diverges ) ) For aroimately what values of ca si be relaced by - with a error of magitude o greater tha -? <. <. <.7 <.9 ) Use artial fractios to fid the sum of the series. ) /(+) - / The series diverges. - - ) Fid the Taylor olyomial of lowest degree that will aroimate F() throughout the give iterval with a error of magitude less tha tha -. 9) F() = e-t dt, [,.] 9)