An alternating series is a series where the signs alternate. Generally (but not always) there is a factor of the form ( 1) n + 1
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1 Calculus II - Problem Solvig Drill 20: Alteratig Series, Ratio ad Root Tests Questio No. of 0 Istructios: () Read the problem ad aswer choices carefully (2) Work the problems o paper as eeded (3) Pick the aswer (4) Go back to review the core cocept tutorial as eeded.. A alteratig series meas: Questio #0 (A) The series alterates betwee covergece ad divergece. (B) The terms of the series alterate sigs. (C) The series coverges. (D) The series alterates betwee geometric ad the p-series. (E) The series diverges. A. Icorrect! It has to do with the sigs of the terms. B. Correct! This is the correct aswer. A alteratig series has terms that alterate i sigs. Feedback o Each Aswer C. Icorrect! It has to do with the sigs of the terms. D. Icorrect! It has to do with the sigs of the terms. (E) Icorrect! It has to do with the sigs of the terms. A alteratig series is a series where the sigs alterate. Geerally (but ot always) there is a factor of the form ( ) + or ( ) i the series. Solutio RapidLearigCeter.com Rapid Learig Ic. All Rights Reserved
2 Questio No. 2 of 0 Istructios: () Read the problem ad aswer choices carefully (2) Work the problems o paper as eeded (3) Pick the aswer (4) Go back to review the core cocept tutorial as eeded. 2. If a >0 ad lim a = 0, the the Alteratig Series Test states that coverge if which of the followig coditios are met? ( ) a will Questio #02 (A) a + = (B) a + (C) + a a a a (D) + = a a (E) a + a A. Icorrect Remember the coditios of the Alteratig Series Test. B. Icorrect! Remember the coditios of the Alteratig Series Test. Feedback o Each Aswer C. Icorrect! Remember the coditios of the Alteratig Series Test. D. Icorrect! Remember the coditios of the Alteratig Series Test. E. Correct! This is the correct aswer. This idicates that a[] is oicreasig. Recall that the Alteratig Series Test states that if a oicreasig (i.e., a + a ), the ( ) a coverges. >0 ad if a ad a is lim = 0 Solutio RapidLearigCeter.com Rapid Learig Ic. All Rights Reserved
3 Questio No. 3 of 0 Istructios: () Read the problem ad aswer choices carefully (2) Work the problems o paper as eeded (3) Pick the aswer (4) Go back to review the core cocept tutorial as eeded. 3. The differece betwee ( ) a ad N ( ) a for some N is called the: Questio #03 (A) Term (B) Sum (C) Harmoic (D) Remaider (E) Covergece A. Icorrect! The differece is ot called the term. B. Icorrect! The sum is the value of the etire series. Feedback o Each Aswer C. Icorrect! A harmoic series has terms i the form /. D. Correct! This is the correct aswer. The remaider is the differece betwee the ifiite series ad the partial sum. E. Icorrect! The covergece tells whether the series sums to a fiite umber. The differece betwee ( ) a ad N ( ) a for some N is called the remaider. The remaider is useful whe approximatig a series by a partial sum. Solutio RapidLearigCeter.com Rapid Learig Ic. All Rights Reserved
4 Questio No. 4 of 0 Istructios: () Read the problem ad aswer choices carefully (2) Work the problems o paper as eeded (3) Pick the aswer (4) Go back to review the core cocept tutorial as eeded. 4. Which of the followig is ot a alteratig series? Questio #04 (A) (B) (C) (D) (E) 2 ( ) + 2 ( ) ( ) ( ) ( ) A. Icorrect! This is a alteratig series. B. Icorrect! This is a alteratig series. Feedback o Each Aswer C. Correct! This is the correct aswer. Try writig out the first few terms. D. Icorrect! This is a alteratig series. E. Icorrect! This is a alteratig series. By writig the first few terms, it is easily see that ( ) 2 2 is the oly series with all positive terms. This is because 2 is eve for ay therefore 2 ( ) is always positive. Solutio RapidLearigCeter.com Rapid Learig Ic. All Rights Reserved
5 Questio No. 5 of 0 Istructios: () Read the problem ad aswer choices carefully (2) Work the problems o paper as eeded (3) Pick the aswer (4) Go back to review the core cocept tutorial as eeded. 5. Give the series ( ), if we were to estimate the sum usig the first 4 terms of the sequece, the we could coclude that the remaider R: Questio #05 (A) Is greater tha 0.2. (B) Equals 0.2. (C) Is less that (D) Is less tha 0.2. (E) Is greater tha A. Icorrect! Remember that the remaider is less tha a +. B. Icorrect! Remember that the remaider is less tha a +. Feedback o Each Aswer C. Icorrect! Remember that the remaider is less tha a +. D. Correct! This the correct aswer. The remaider is less tha a +. E. Icorrect! Remember that the remaider is less tha a + Give that ( ) remaider is less tha 5 is estimated usig the first four terms of the series, the a = = Solutio RapidLearigCeter.com Rapid Learig Ic. All Rights Reserved
6 Questio No. 6 of 0 Istructios: () Read the problem statemet ad aswer choices carefully (2) Work the problems o paper as eeded (3) Pick the aswer (4) Go back to review the core cocept tutorial as eeded. 6. If a series coverges absolutely, the it. Questio #06 (A) May diverge. (B) Must be a alteratig series. (C) Must be a geometric series. (D) Must be a p-series. (E) Is guarateed to coverge. A. Icorrect! Recall the relatioship betwee absolute covergece ad geeral covergece. B. Icorrect! Recall the relatioship betwee absolute covergece ad geeral covergece. Feedback o Each Aswer choice C. Icorrect! Recall the relatioship betwee absolute covergece ad geeral covergece. D. Icorrect! Recall the relatioship betwee absolute covergece ad geeral covergece. E. Correct! This the correct aswer. If a series coverges absolutely, the it geerally coverges. If a series coverges absolutely, the it geerally coverges. Note that the coverse is ot true. Solutio RapidLearigCeter.com Rapid Learig Ic. All Rights Reserved
7 Questio No. 7 of 0 Istructios: () Read the problem ad aswer choices carefully (2) Work the problems o paper as eeded (3) Pick the aswer (4) Go back to review the core cocept tutorial as eeded. 7. The series /2 ( ). Questio #07 (A) Coverges absolutely. (B) Coverges coditioally. (C) Diverges. (D) Is a harmoic series. (E) Noe of these. A. Icorrect! 7 Compare the covergece of the series with the covergece of the series of the absolute value of each term. B. Correct! This is the correct aswer. The series coverges coditioally. Feedback o Each Aswer C. Icorrect! Compare the covergece of the series with the covergece of the series of the absolute value of each term. D. Icorrect! The series is ot harmoic. E. Icorrect! Oe of the give aswers is correct, please try agai. By the alteratig series test, /2 ( ) = /2 coverges coditioally. /2 ( ) coverges. However, which is a diverget p-series. Therefore, /2 ( ) Solutio RapidLearigCeter.com Rapid Learig Ic. All Rights Reserved
8 Questio No. 8 of 0 Istructios: () Read the problem ad aswer choices carefully (2) Work the problems o paper as eeded (3) Pick the aswer (4) Go back to review the core cocept tutorial as eeded. 8. Usig the Ratio Test for the series 5! coclusio ca be made about the covergece?, what is the limitig value ad what Questio #08 (A) 0; Diverges (B) 0; Coverges (C) 5; Coverges (D) 5; Diverges (E) ; Icoclusive A. Icorrect! The limitig value is correct, but the series coverges. B. Correct! This is the correct aswer. The series coverges by the Ratio Test. A Feedback o Each Aswer C. Icorrect! The limitig value is icorrect. D. Icorrect! The limitig value is icorrect. E. Icorrect! The limitig value is icorrect. Give the series 5!, we use the Ratio Test to fid + 5 ( + )! 5 lim = lim = 0 5 +! The Ratio Test states that if the limitig value is less tha, the series coverges.. Solutio RapidLearigCeter.com Rapid Learig Ic. All Rights Reserved
9 Questio No. 9 of 0 Istructios: () Read the problem ad aswer choices carefully (2) Work the problems o paper as eeded (3) Pick the aswer (4) Go back to review the core cocept tutorial as eeded. 9. Usig the Ratio Test for the series! coclusio ca be made about the covergece?, what is the limitig value ad what Questio #09 (A) e; Diverges (B) e; Coverges (C) ; Coverges (D) ; Diverges (E) ; Icoclusive A. Correct! This is the correct aswer. The series coverges by the Ratio Test. B. Icorrect! The limitig value is correct, but the series diverges. A Feedback o Each Aswer C. Icorrect! The limitig value is icorrect. D. Icorrect! The limitig value is icorrect. E. Icorrect! The limitig value is icorrect. Give the series!, we use the Ratio Test to fid + ( + )! ( + ) lim = lim = e! The Ratio Test states that if the limitig value is greater tha, the series diverges.. Solutio RapidLearigCeter.com Rapid Learig Ic. All Rights Reserved
10 Questio No. 0 of 0 Istructios: () Read the problem ad aswer choices carefully (2) Work the problems o paper as eeded (3) Pick the aswer (4) Go back to review the core cocept tutorial as eeded. 0. Usig the Root Test for the series 2 coclusio ca be made about the covergece?, what is the limitig value ad what Questio #0 (A) ; Diverges (B) ; Coverges (C) 2; Coverges (D) ; Diverges (E) 2; Diverges A. Icorrect! The limitig value is icorrect. B. Icorrect! The limitig value is icorrect. Feedback o Each Aswer C. Icorrect! The limitig value is correct, but the series diverges. D. Icorrect! The limitig value is icorrect. E. Correct! This is the correct aswer. The series diverges by the Root Test. Give the series 2 the Root Test, the series diverges. 2 2 lim = lim = 2 /, we use the Root Test to fid. By Solutio RapidLearigCeter.com Rapid Learig Ic. All Rights Reserved
E. Incorrect! Plug n = 1, 2, 3, & 4 into the general term formula. n =, then the first four terms are found by
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