Calculus II - Problem Solvig Drill 8: Sequeces, Series, ad Covergece Questio No. of 0. Fid the first four terms of the sequece whose geeral term is give by a ( ) : Questio #0 (A) (B) (C) (D) (E) 8,,, 4 8,,, 4 8,,, 4 8,,, 4 8 4,,, A. Icorrect! The sigs are the terms are icorrect. B. Correct! This is the correct aswer foud by pluggig,,, & 4 ito the geeral formula. Feedback o Each Aswer C. Icorrect! The sigs are the terms are icorrect. D. Icorrect! The sigs are the terms are icorrect. E. Icorrect! Plug,,, & 4 ito the geeral term formula. If the geeral term is give by a ( ), the the first four terms are foud by 8 pluggig,,, & 4 ito the formula. We have a, a, a, ad 4 4 a. Solutio RapidLearigCeter.com Rapid Learig Ic. All Rights Reserved
Questio No. of 0. Fid the first four terms of the sequece whose geeral term is give by a ( ) : Questio #0 (A),, 9, 64 (B),, 9, 64 (C),, 9, 64 (D),, 9, 64 (E),, 9, 64 A. Correct This is the correct aswer foud by pluggig,,, & 4 ito the geeral formula. B. Icorrect! The sigs are the terms are icorrect. Feedback o Each Aswer C. Icorrect! The sigs are the terms are icorrect. D. Icorrect! The sigs are the terms are icorrect. E. Icorrect! The sigs are the terms are icorrect. If the geeral term is give by a ( ), the the first four terms are foud by pluggig,,, & 4 ito the formula. We have a, a, a 9, ad 4 64 a. Solutio RapidLearigCeter.com Rapid Learig Ic. All Rights Reserved
Questio No. of 0. Fid a geeral formula for the sequece 4 8,,,,... 5 Questio #0 (A) (B) (C) (D) a a a + + a (E) Ca t be determied A. Icorrect! Try to recogize a patter. It might help to rewrite / as 6/4. B. Icorrect! Try to recogize a patter. It might help to rewrite / as 6/4. Feedback o Each Aswer C. Correct! This is the correct aswer. It is foud by recogizig the patter. D. Icorrect! Try to recogize a patter. It might help to rewrite / as 6/4. E. Icorrect! Oe of the give aswers is correct. Please try agai. We begi by rewritig the sequece 4 8,,,,... 5 as 4 6 8,,,,... 4 5 We may ow recogize the umerator as ad the deomiator as +. Therefore, the geeral term is a. + Solutio RapidLearigCeter.com Rapid Learig Ic. All Rights Reserved
Questio No. 4 of 0 4. Cosider the sequece with geeral term sequece coverge to? a +. What value does this Questio #04 (A) + (B) (C) (D) (E) + A. Icorrect! Take the limit as approaches ifiity. B. Icorrect! Take the limit as approaches ifiity. Feedback o Each Aswer C. Correct! This is the correct aswer. It is foud by takig the limit as approaches ifiity. D. Icorrect! Take the limit as approaches ifiity. E. Icorrect! Take the limit as approaches ifiity. a + lim a lim. + We fid the value that We have coverges to by takig the limit as approaches. Solutio RapidLearigCeter.com Rapid Learig Ic. All Rights Reserved
Questio No. 5 of 0 5. The sequece with geeral term a is: Questio #05 (A) Bouded (B) Mootoic (C) Bouded ad mootoic (D) Not mootoic ad ot bouded (E) Noe of these. A. Icorrect! The sequece is ot bouded. B. Correct! This is the correct aswer. The sequece is mootoic, but ot bouded. Feedback o Each Aswer C. Icorrect! The sequece is ot bouded. D. Icorrect! The sequece is mootoic. E. Icorrect! Oe of the give choices is correct. Please try agai. Give a, it is clear that the sequece is ot bouded sice approaches as gets larger ad larger. Sice this is a strictly icreasig sequece, it is mootoic. Solutio RapidLearigCeter.com Rapid Learig Ic. All Rights Reserved
Questio No. 6 of 0 6. a is called Questio #06 (A) A ifiite series (B) A ifiite sequece (C) A geeral term (D) A fiite series (E) A fiite sequece A. Correct! This is the correct aswer. This is a ifiite series. B. Icorrect! A sequece is a list of terms, ot the sum. Feedback o Each Aswer choice C. Icorrect! The geeral term is part of this expressio, but the overall expressio is a ifiite series. D. Icorrect! A fiite series would ot have ifiity as the upper limit. E. Icorrect! A sequece is a list of terms, ot the sum. Expressios i the form a are called ifiite series. Solutio RapidLearigCeter.com Rapid Learig Ic. All Rights Reserved
Questio No. 7 of 0 7. What coclusio ca be made about the covergece or divergece of usig the divergece test? Questio #07 (A) Coverge (B) Both coverge ad diverge (C) Diverge (D) Neither coverge or diverge (E) No coclusio ca be made A. Icorrect! Take the limit of the geeral term a approaches ifiity. B. Icorrect! It is ot possible to both coverge ad diverge at the same time. Feedback o Each Aswer C. Correct! This is the correct aswer. Sice the limit of () as approaches ifiity is ot zero, the series diverges. D. Icorrect! The series must either coverge or diverge. E. Icorrect! Take the limit of the geeral term a approaches ifiity. Give, we fid that lim( ). By the divergece test, the series diverges. Solutio RapidLearigCeter.com Rapid Learig Ic. All Rights Reserved
Questio No. 8 of 0 8. What coclusio ca be made about the covergece or divergece of usig the divergece test? Questio #08 (A) Coverge (B) Both coverge ad diverge (C) Diverge (D) Neither coverge or diverge (E) No coclusio ca be made A. Icorrect! The series does ot coverge. Remember that just because / has a limitig value of 0, this does ot guaratee covergece. B. Icorrect! It is ot possible to both coverge ad diverge at the same time. A Feedback o Each Aswer C. Icorrect! The series actually does diverge. However, this caot be determied based o the divergece test. D. Icorrect! The series must either coverge or diverge. E. Correct! This is the correct aswer. Based o this tutorial, eve though the limitig value of / is 0, o coclusio ca be made about the covergece without further aalysis. Give, we have lim 0. However, this does ot guaratee covergece accordig to the divergece test. Therefore, o coclusio ca be made based o the divergece test. This series is actually a special case of a ifiite series called the Harmoic series ad is kow to diverge. Solutio RapidLearigCeter.com Rapid Learig Ic. All Rights Reserved
Questio No. 9 of 0 9. Fid the value of (A) i 0 i Questio #09 (B) (C) (D) (E) The series does ot coverge A. Correct! This is the correct aswer. Sice the series is a coverget geometric series, its sum ca be foud easily. B. Icorrect! The series is a covergig geometric series. A Feedback o Each Aswer C. Icorrect! The series is a covergig geometric series. D. Icorrect! The series is a covergig geometric series. E. Icorrect! The series does coverge. It is a geometric series. Note that i 0 i is a geometric series with a ad a r that the series coverges to. r. Sice < r <, we kow Solutio RapidLearigCeter.com Rapid Learig Ic. All Rights Reserved
Questio No. 0 of 0 Questio #0 0. Fid the value of (A) 5 (B) 0 (C) i 0 i 5 (D) 5 (E) The series does ot coverge A. Icorrect! The series is a covergig geometric series. Feedback o Each Aswer B. Correct! This is the correct aswer. Sice the series is a coverget geometric series, its sum ca be foud easily. C. Icorrect! The series is a covergig geometric series. D. Icorrect! The series is a covergig geometric series. E. Icorrect! The series does coverge. It is a geometric series. Note that i 0 i 5 is a geometric series with a 5 ad a 5 0 r kow that the series coverges to. r. Sice < r <, we Solutio RapidLearigCeter.com Rapid Learig Ic. All Rights Reserved