722.6 693.1 833.5 795.8 382.6 545.5 825.4 663.6 972.9 795.8 826.4 722.6 826.4 781.6 !A1axw)}p]WgxmkFftu May 3rd, 2018 - Sequences and Series Practice Test Determine if the sequence is arithmetic Find the term named in the problem 27 4 8 16 Sequences and Series Practice for Test Mr C Miller April 30th, 2018 - Determine if the sequence is arithmetic or geometric the problem 3 Sequences and Series Practice for Test Series Algebra II Math Khan Academy Which one of these sequences is a finite sequence? The book contains eight practice tests five practice tests for Calculus AB and three practice tests for Calculus BC. Ex 11.1.2 Use the squeeze theorem to show that \(\lim_{n\to\infty} {n!\over n^n}=0\). Ex 11.5.1 \(\sum_{n=1}^\infty {1\over 2n^2+3n+5} \) (answer), Ex 11.5.2 \(\sum_{n=2}^\infty {1\over 2n^2+3n-5} \) (answer), Ex 11.5.3 \(\sum_{n=1}^\infty {1\over 2n^2-3n-5} \) (answer), Ex 11.5.4 \(\sum_{n=1}^\infty {3n+4\over 2n^2+3n+5} \) (answer), Ex 11.5.5 \(\sum_{n=1}^\infty {3n^2+4\over 2n^2+3n+5} \) (answer), Ex 11.5.6 \(\sum_{n=1}^\infty {\ln n\over n}\) (answer), Ex 11.5.7 \(\sum_{n=1}^\infty {\ln n\over n^3}\) (answer), Ex 11.5.8 \(\sum_{n=2}^\infty {1\over \ln n}\) (answer), Ex 11.5.9 \(\sum_{n=1}^\infty {3^n\over 2^n+5^n}\) (answer), Ex 11.5.10 \(\sum_{n=1}^\infty {3^n\over 2^n+3^n}\) (answer). Khan Academy is a 501(c)(3) nonprofit organization. To log in and use all the features of Khan Academy, please enable JavaScript in your browser. 1 2 + 1 4 + 1 8 + = n=1 1 2n = 1 We will need to be careful, but it turns out that we can . << 772.4 811.3 431.9 541.2 833 666.2 947.3 784.1 748.3 631.1 775.5 745.3 602.2 573.9 /BaseFont/BPHBTR+CMMI12 stream We will also illustrate how the Ratio Test and Root Test can be used to determine the radius and interval of convergence for a power series. /Subtype/Type1 Learning Objectives. /BaseFont/CQGOFL+CMSY10 The Alternating Series Test can be used only if the terms of the Ex 11.7.5 \(\sum_{n=0}^\infty (-1)^{n}{3^n\over 5^n}\) (answer), Ex 11.7.6 \(\sum_{n=1}^\infty {n!\over n^n}\) (answer), Ex 11.7.7 \(\sum_{n=1}^\infty {n^5\over n^n}\) (answer), Ex 11.7.8 \(\sum_{n=1}^\infty {(n! Psychological Research & Experimental Design, All Teacher Certification Test Prep Courses. Published by Wiley. endstream stream 531.3 590.3 560.8 414.1 419.1 413.2 590.3 560.8 767.4 560.8 560.8 472.2 531.3 1062.5 << Comparison Test: This applies . For problems 1 - 3 perform an index shift so that the series starts at n = 3 n = 3. Therefore the radius of convergence is R = , and the interval of convergence is ( - , ). 8 0 obj To log in and use all the features of Khan Academy, please enable JavaScript in your browser. )^2\over n^n}(x-2)^n\) (answer), Ex 11.8.6 \(\sum_{n=1}^\infty {(x+5)^n\over n(n+1)}\) (answer), Ex 11.9.1 Find a series representation for \(\ln 2\). /Subtype/Type1 Proofs for both tests are also given. 413.2 531.3 826.4 295.1 354.2 295.1 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 Here are a set of practice problems for the Series and Sequences chapter of the Calculus II notes. Legal. Boundary Value Problems & Fourier Series, 8.3 Periodic Functions & Orthogonal Functions, 9.6 Heat Equation with Non-Zero Temperature Boundaries, 1.14 Absolute Value Equations and Inequalities. (answer), Ex 11.2.8 Compute \(\sum_{n=1}^\infty \left({3\over 5}\right)^n\). }\) (answer), Ex 11.8.3 \(\sum_{n=1}^\infty {n!\over n^n}x^n\) (answer), Ex 11.8.4 \(\sum_{n=1}^\infty {n!\over n^n}(x-2)^n\) (answer), Ex 11.8.5 \(\sum_{n=1}^\infty {(n! endstream endobj 208 0 obj <. If you'd like a pdf document containing the solutions the download tab above contains links to pdf's containing the solutions for the full book, chapter and section. Math C185: Calculus II (Tran) 6: Sequences and Series 6.5: Comparison Tests 6.5E: Exercises for Comparison Test Expand/collapse global location 6.5E: Exercises for Comparison Test . endobj /BaseFont/UNJAYZ+CMR12 /Length 465 All other trademarks and copyrights are the property of their respective owners. 68 0 obj hb```9B 7N0$K3 }M[&=cx`c$Y&a YG&lwG=YZ}w{l;r9P"J,Zr]Ngc E4OY%8-|\C\lVn@`^) E 3iL`h`` !f s9B`)qLa0$FQLN$"H&8001a2e*9y,Xs~z1111)QSEJU^|2n[\\5ww0EHauC8Gt%Y>2@ " Worked example: sequence convergence/divergence, Partial sums: formula for nth term from partial sum, Partial sums: term value from partial sum, Worked example: convergent geometric series, Worked example: divergent geometric series, Infinite geometric series word problem: bouncing ball, Infinite geometric series word problem: repeating decimal, Proof of infinite geometric series formula, Level up on the above skills and collect up to 320 Mastery points, Determine absolute or conditional convergence, Level up on the above skills and collect up to 640 Mastery points, Worked example: alternating series remainder, Taylor & Maclaurin polynomials intro (part 1), Taylor & Maclaurin polynomials intro (part 2), Worked example: coefficient in Maclaurin polynomial, Worked example: coefficient in Taylor polynomial, Visualizing Taylor polynomial approximations, Worked example: estimating sin(0.4) using Lagrange error bound, Worked example: estimating e using Lagrange error bound, Worked example: cosine function from power series, Worked example: recognizing function from Taylor series, Maclaurin series of sin(x), cos(x), and e, Finding function from power series by integrating, Interval of convergence for derivative and integral, Integrals & derivatives of functions with known power series, Formal definition for limit of a sequence, Proving a sequence converges using the formal definition, Infinite geometric series formula intuition, Proof of infinite geometric series as a limit. When you have completed the free practice test, click 'View Results' to see your results. Which of the following represents the distance the ball bounces from the first to the seventh bounce with sigma notation? . >> /FontDescriptor 11 0 R nn = 0. (answer). 826.4 531.3 958.7 1076.8 826.4 295.1 295.1 531.3 885.4 531.3 885.4 826.4 295.1 413.2 Premium members get access to this practice exam along with our entire library of lessons taught by subject matter experts. Part II. A proof of the Alternating Series Test is also given. Strip out the first 3 terms from the series n=1 2n n2 +1 n = 1 2 n n 2 + 1. All rights reserved. Learn for free about math, art, computer programming, economics, physics, chemistry, biology, medicine, finance, history, and more. Indiana Core Assessments Mathematics: Test Prep & Study Guide. If you'd like a pdf document containing the solutions the download tab above contains links to pdf's containing the solutions for the full book, chapter and section. 326.4 272 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 272 272 The Root Test can be used on any series, but unfortunately will not always yield a conclusive answer as to whether a series will converge absolutely or diverge. /LastChar 127 A review of all series tests. /Type/Font In other words, a series is the sum of a sequence. ]^e-V!2 F. /Length 569 /Name/F1 2 6 points 2. /Widths[777.8 277.8 777.8 500 777.8 500 777.8 777.8 777.8 777.8 777.8 777.8 777.8 1. 762 689.7 1200.9 820.5 796.1 695.6 816.7 847.5 605.6 544.6 625.8 612.8 987.8 713.3 (answer), Ex 11.3.12 Find an \(N\) so that \(\sum_{n=2}^\infty {1\over n(\ln n)^2}\) is between \(\sum_{n=2}^N {1\over n(\ln n)^2}\) and \(\sum_{n=2}^N {1\over n(\ln n)^2} + 0.005\). Then determine if the series converges or diverges. Maclaurin series of e, sin(x), and cos(x). Ex 11.1.1 Compute \(\lim_{x\to\infty} x^{1/x}\). Ex 11.10.8 Find the first four terms of the Maclaurin series for \(\tan x\) (up to and including the \( x^3\) term). )^2\over n^n}\) (answer). We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. x[[o6~cX/e`ElRm'1%J$%v)tb]1U2sRV}.l%s\Y UD+q}O+J /Name/F3 /Filter /FlateDecode A proof of the Root Test is also given. /Type/Font Calculus II-Sequences and Series. endobj << << Some infinite series converge to a finite value. (answer), Ex 11.2.2 Explain why \(\sum_{n=1}^\infty {5\over 2^{1/n}+14}\) diverges. << (answer). /Type/Font When you have completed the free practice test, click 'View Results' to see your results. >> Alternating Series Test - In this section we will discuss using the Alternating Series Test to determine if an infinite series converges or diverges. Here are a set of practice problems for the Series and Sequences chapter of the Calculus II notes. However, use of this formula does quickly illustrate how functions can be represented as a power series. Comparison Test/Limit Comparison Test In this section we will discuss using the Comparison Test and Limit Comparison Tests to determine if an infinite series converges or diverges. At this time, I do not offer pdf's for . Ratio test. Ex 11.1.3 Determine whether {n + 47 n} . 1277.8 555.6 1000 1444.4 555.6 1000 1444.4 472.2 472.2 527.8 527.8 527.8 527.8 666.7 hbbd```b``~"A$" "Y`L6`RL,-`sA$w64= f[" RLMu;@jAl[`3H^Ne`?$4 (answer), Ex 11.1.4 Determine whether \(\left\{{n^2+1\over (n+1)^2}\right\}_{n=0}^{\infty}\) converges or diverges. About this unit. bmkraft7. Learn how this is possible, how we can tell whether a series converges, and how we can explore convergence in . xTn0+,ITi](N@ fH2}W"UG'.% Z#>y{!9kJ+ 508.8 453.8 482.6 468.9 563.7 334 405.1 509.3 291.7 856.5 584.5 470.7 491.4 434.1 Complementary General calculus exercises can be found for other Textmaps and can be accessed here. What is the radius of convergence? /Filter[/FlateDecode] Premium members get access to this practice exam along with our entire library of lessons taught by subject matter experts. 326.4 272 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 272 272 )Ltgx?^eaT'&+n+hN4*D^UR!8UY@>LqS%@Cp/-12##DR}miBw6"ja+WjU${IH$5j!j-I1 The Ratio Test can be used on any series, but unfortunately will not always yield a conclusive answer as to whether a series will converge absolutely or diverge. /Filter /FlateDecode If you're seeing this message, it means we're having trouble loading external resources on our website. If youd like to view the solutions on the web go to the problem set web page, click the solution link for any problem and it will take you to the solution to that problem. Ex 11.1.3 Determine whether \(\{\sqrt{n+47}-\sqrt{n}\}_{n=0}^{\infty}\) converges or diverges. Final: all from 02/05 and 03/11 exams (except work, separation of variables, and probability) plus sequences, series, convergence tests, power series, Taylor series. >> Series The Basics In this section we will formally define an infinite series. /Widths[611.8 816 761.6 679.6 652.8 734 707.2 761.6 707.2 761.6 707.2 571.2 544 544 /LastChar 127 Which is the infinite sequence starting with 1 where each number is the previous number times 3? 979.2 979.2 979.2 272 272 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 Bottom line -- series are just a lot of numbers added together. A brick wall has 60 bricks in the first row, but each row has 3 fewer bricks than the previous one. 2.(a). Which of the following is the 14th term of the sequence below? 21 terms. Section 10.3 : Series - Basics. Differentiate u to find du, and integrate dv to find v. Use the formula: Evaluate the right side of this equation to solve the integral. Applications of Series In this section we will take a quick look at a couple of applications of series. Sequences can be thought of as functions whose domain is the set of integers. We will also briefly discuss how to determine if an infinite series will converge or diverge (a more in depth discussion of this topic will occur in the next section). 62 0 obj Then click 'Next Question' to answer the next question. Other sets by this creator. Infinite series are sums of an infinite number of terms. Power Series and Functions In this section we discuss how the formula for a convergent Geometric Series can be used to represent some functions as power series. Quiz 2: 8 questions Practice what you've learned, and level up on the above skills. In addition, when \(n\) is not an integer an extension to the Binomial Theorem can be used to give a power series representation of the term. 833.3 833.3 833.3 833.3 1444.4 1277.8 555.6 1111.1 1111.1 1111.1 1111.1 1111.1 944.4 Calculus 2. Choose your answer to the question and click 'Continue' to see how you did. xWKoFWlojCpP NDED$(lq"g|3g6X_&F1BXIM5d gOwaN9c,r|9 590.3 767.4 795.8 795.8 1091 795.8 795.8 649.3 295.1 531.3 295.1 531.3 295.1 295.1 The following is a list of worksheets and other materials related to Math 129 at the UA. /Length 2492 18 0 obj endobj If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. Contact us by phone at (877)266-4919, or by mail at 100ViewStreet#202, MountainView, CA94041. (answer), Ex 11.9.2 Find a power series representation for \(1/(1-x)^2\). 1000 1000 1000 777.8 275 1000 666.7 666.7 888.9 888.9 0 0 555.6 555.6 666.7 500 722.2 In order to use either test the terms of the infinite series must be positive. OR sequences are lists of numbers, where the numbers may or may not be determined by a pattern. (answer), Ex 11.2.5 Compute \(\sum_{n=0}^\infty {3\over 2^n}+ {4\over 5^n}\). /LastChar 127 665 570.8 924.4 812.6 568.1 670.2 380.8 380.8 380.8 979.2 979.2 410.9 514 416.3 421.4 Which of the following is the 14th term of the sequence below? 888.9 888.9 888.9 888.9 888.9 888.9 888.9 666.7 875 875 875 875 611.1 611.1 833.3 Some infinite series converge to a finite value. /BaseFont/VMQJJE+CMR8 Remark. Choose your answer to the question and click 'Continue' to see how you did. /Name/F5 Contact us by phone at (877)266-4919, or by mail at 100ViewStreet#202, MountainView, CA94041. /Name/F4 MULTIPLE CHOICE: Circle the best answer. 531.3 531.3 531.3 295.1 295.1 295.1 826.4 501.7 501.7 826.4 795.8 752.1 767.4 811.1 Infinite series are sums of an infinite number of terms. All rights reserved. |: The Ratio Test shows us that regardless of the choice of x, the series converges. When you have completed the free practice test, click 'View Results' to see your results. n = 1 n2 + 2n n3 + 3n2 + 1. Ex 11.4.1 \(\sum_{n=1}^\infty {(-1)^{n-1}\over 2n+5}\) (answer), Ex 11.4.2 \(\sum_{n=4}^\infty {(-1)^{n-1}\over \sqrt{n-3}}\) (answer), Ex 11.4.3 \(\sum_{n=1}^\infty (-1)^{n-1}{n\over 3n-2}\) (answer), Ex 11.4.4 \(\sum_{n=1}^\infty (-1)^{n-1}{\ln n\over n}\) (answer), Ex 11.4.5 Approximate \(\sum_{n=1}^\infty (-1)^{n-1}{1\over n^3}\) to two decimal places. We use the geometric, p-series, telescoping series, nth term test, integral test, direct comparison, limit comparison, ratio test, root test, alternating series test, and the test. Note as well that there really isnt one set of guidelines that will always work and so you always need to be flexible in following this set of guidelines. For problems 1 3 perform an index shift so that the series starts at \(n = 3\). << Alternating Series Test In this section we will discuss using the Alternating Series Test to determine if an infinite series converges or diverges. Integral test. Khan Academy is a nonprofit with the mission of providing a free, world-class education for anyone, anywhere. Ex 11.7.4 Compute \(\lim_{n\to\infty} |a_n|^{1/n}\) for the series \(\sum 1/n\). We will also determine a sequence is bounded below, bounded above and/or bounded. Worksheets. Find the sum of the following geometric series: The formula for a finite geometric series is: Which of these is an infinite sequence of all the non-zero even numbers beginning at number 2? The test was used by Gottfried Leibniz and is sometimes known as Leibniz's test, Leibniz's rule, or the Leibniz criterion.The test is only sufficient, not necessary, so some convergent . Which of the following sequences is NOT a geometric sequence? Martha_Austin Teacher. Khan Academy is a 501(c)(3) nonprofit organization. Choose the equation below that represents the rule for the nth term of the following geometric sequence: 128, 64, 32, 16, 8, . We will also see how we can use the first few terms of a power series to approximate a function. (1 point) Is the integral Z 1 1 1 x2 dx an improper integral? Below are some general cases in which each test may help: P-Series Test: The series be written in the form: P 1 np Geometric Series Test: When the series can be written in the form: P a nrn1 or P a nrn Direct Comparison Test: When the given series, a 252 0 obj <>stream Sequences In this section we define just what we mean by sequence in a math class and give the basic notation we will use with them. /Widths[458.3 458.3 416.7 416.7 472.2 472.2 472.2 472.2 583.3 583.3 472.2 472.2 333.3 \ _* %l~G"tytO(J*l+X@ uE: m/ ~&Q24Nss(7F!ky=4 Mijo8t;v % Find the radius and interval of convergence for each of the following series: Solution (a) We apply the Ratio Test to the series n = 0 | x n n! Free Practice Test Instructions: Choose your answer to the question and click 'Continue' to see how you did. UcTIjeB#vog-TM'FaTzG(:k-BNQmbj}'?^h<=XgS/]o4Ilv%Jm 979.2 489.6 489.6 489.6] << A proof of the Ratio Test is also given. 272 761.6 462.4 462.4 761.6 734 693.4 707.2 747.8 666.2 639 768.3 734 353.2 503 761.2 L7s[AQmT*Z;HK%H0yqt1r8 endstream Our mission is to provide a free, world-class education to anyone, anywhere. Comparison tests. It turns out the answer is no. copyright 2003-2023 Study.com. 21 0 obj Which is the finite sequence of four multiples of 9, starting with 9? 556.5 425.2 527.8 579.5 613.4 636.6 609.7 458.2 577.1 808.9 505 354.2 641.4 979.2 stream 5.3.3 Estimate the value of a series by finding bounds on its remainder term. Choose your answer to the question and click 'Continue' to see how you did. endobj n = 1 n 2 + 2 n n 3 + 3 n . 500 388.9 388.9 277.8 500 500 611.1 500 277.8 833.3 750 833.3 416.7 666.7 666.7 777.8 Our mission is to provide a free, world-class education to anyone, anywhere. /Filter /FlateDecode endobj For each of the following series, determine which convergence test is the best to use and explain why. Strategy for Series In this section we give a general set of guidelines for determining which test to use in determining if an infinite series will converge or diverge. Then we can say that the series diverges without having to do any extra work. 816 816 272 299.2 489.6 489.6 489.6 489.6 489.6 734 435.2 489.6 707.2 761.6 489.6 24 0 obj (answer), Ex 11.2.1 Explain why \(\sum_{n=1}^\infty {n^2\over 2n^2+1}\) diverges. 507.9 433.7 395.4 427.7 483.1 456.3 346.1 563.7 571.2 589.1 483.8 427.7 555.4 505 979.2 489.6 489.6 489.6] /Length 1722 17 0 obj Learn how this is possible, how we can tell whether a series converges, and how we can explore convergence in Taylor and Maclaurin series. (answer), Ex 11.9.3 Find a power series representation for \( 2/(1-x)^3\). We also discuss differentiation and integration of power series. /Name/F2 Special Series In this section we will look at three series that either show up regularly or have some nice properties that we wish to discuss. Complementary General calculus exercises can be found for other Textmaps and can be accessed here. If it con-verges, nd the limit. With an outline format that facilitates quick and easy review, Schaum's Outline of Calculus, Seventh Edition helps you understand basic concepts and get the extra practice you need to excel in these courses. %PDF-1.5 Root Test In this section we will discuss using the Root Test to determine if an infinite series converges absolutely or diverges. stream (answer), Ex 11.2.3 Explain why \(\sum_{n=1}^\infty {3\over n}\) diverges. Here are a set of practice problems for the Series and Sequences chapter of the Calculus II notes. 531.3 531.3 531.3] 666.7 1000 1000 1000 1000 1055.6 1055.6 1055.6 777.8 666.7 666.7 450 450 450 450 272 816 544 489.6 544 516.8 380.8 386.2 380.8 544 516.8 707.2 516.8 516.8 435.2 489.6 << Psychological Research & Experimental Design, All Teacher Certification Test Prep Courses. 1 2, 1 4, 1 8, Sequences of values of this type is the topic of this rst section. Sequences & Series in Calculus Chapter Exam. 722.2 777.8 777.8 611.1 798.5 656.8 526.5 771.4 527.8 718.7 594.9 844.5 544.5 677.8 Convergence/Divergence of Series In this section we will discuss in greater detail the convergence and divergence of infinite series. /LastChar 127 >> &/ r YesNo 2.(b). Here are a set of practice problems for the Series and Sequences chapter of the Calculus II notes. Course summary; . Study Online AP Calculus AB and BC: Chapter 9 -Infinite Sequences and Series : 9.2 -The Integral Test and p-Series Study Notes Prepared by AP Teachers Skip to content . Chapters include Linear Math 106 (Calculus II): old exams. A ball is dropped from an unknown height (h) and it repeatedly bounces on the floor. Chapter 10 : Series and Sequences. The Alternating Series Test can be used only if the terms of the series alternate in sign. /Widths[611.8 816 761.6 679.6 652.8 734 707.2 761.6 707.2 761.6 707.2 571.2 544 544 For each function, find the Maclaurin series or Taylor series centered at $a$, and the radius of convergence. The sum of the steps forms an innite series, the topic of Section 10.2 and the rest of Chapter 10. (answer), Ex 11.3.10 Find an \(N\) so that \(\sum_{n=0}^\infty {1\over e^n}\) is between \(\sum_{n=0}^N {1\over e^n}\) and \(\sum_{n=0}^N {1\over e^n} + 10^{-4}\). These are homework exercises to accompany David Guichard's "General Calculus" Textmap. 1) \(\displaystyle \sum^_{n=1}a_n\) where \(a_n=\dfrac{2}{n . Alternating Series Test For series of the form P ( 1)nb n, where b n is a positive and eventually decreasing sequence, then X ( 1)nb n converges ()limb n = 0 POWER SERIES De nitions X1 n=0 c nx n OR X1 n=0 c n(x a) n Radius of convergence: The radius is de ned as the number R such that the power series . 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