Then click 'Next Question' to answer the next question. If it con-verges, nd the limit. /BaseFont/CQGOFL+CMSY10 18 0 obj Choose your answer to the question and click 'Continue' to see how you did. Here are a set of practice problems for the Series and Sequences chapter of the Calculus II notes. n = 1 n2 + 2n n3 + 3n2 + 1. 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 . 777.8 777.8] Sequences can be thought of as functions whose domain is the set of integers. 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 Consider the series n a n. Divergence Test: If lim n a n 0, then n a n diverges. Don't all infinite series grow to infinity? Indiana Core Assessments Mathematics: Test Prep & Study Guide. Then click 'Next Question' to answer the . Khan Academy is a nonprofit with the mission of providing a free, world-class education for anyone, anywhere. /Name/F4 Comparison tests. Donate or volunteer today! My calculus 2 exam on sequence, infinite series & power seriesThe exam: https://bit.ly/36OHYcsAll the convergence tests: https://bit.ly/2IzqokhBest friend an. 883.8 992.6 761.6 272 272 489.6 816 489.6 816 761.6 272 380.8 380.8 489.6 761.6 272 17 0 obj 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 /Length 569 Here are a set of practice problems for the Series and Sequences chapter of the Calculus II notes. Remark. 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. Don't all infinite series grow to infinity? << /Subtype/Type1 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 Ratio Test In this section we will discuss using the Ratio Test to determine if an infinite series converges absolutely or diverges. Math 106 (Calculus II): old exams. /Filter[/FlateDecode] Then click 'Next Question' to answer the next question. /BaseFont/SFGTRF+CMSL12 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 . Question 5 5. /Name/F1 When you have completed the free practice test, click 'View Results' to see your results. Choose the equation below that represents the rule for the nth term of the following geometric sequence: 128, 64, 32, 16, 8, . 611.8 897.2 734 761.6 666.2 761.6 720.6 544 707.2 734 734 1006 734 734 598.4 272 Premium members get access to this practice exam along with our entire library of lessons taught by subject matter experts. Ex 11.7.1 Compute \(\lim_{n\to\infty} |a_{n+1}/a_n|\) for the series \(\sum 1/n^2\). << Ex 11.7.2 Compute \(\lim_{n\to\infty} |a_{n+1}/a_n|\) for the series \(\sum 1/n\). Sequences & Series in Calculus Chapter Exam. 489.6 272 489.6 272 272 489.6 544 435.2 544 435.2 299.2 489.6 544 272 299.2 516.8 1000 1000 1000 777.8 275 1000 666.7 666.7 888.9 888.9 0 0 555.6 555.6 666.7 500 722.2 489.6 272 489.6 272 272 489.6 544 435.2 544 435.2 299.2 489.6 544 272 299.2 516.8 8 0 obj /Widths[663.6 885.4 826.4 736.8 708.3 795.8 767.4 826.4 767.4 826.4 767.4 619.8 590.3 endobj 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 1277.8 555.6 1000 1444.4 555.6 1000 1444.4 472.2 472.2 527.8 527.8 527.8 527.8 666.7 777.8 444.4 444.4 444.4 611.1 777.8 777.8 777.8 777.8] 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 Sequences and Series. 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 Psychological Research & Experimental Design, All Teacher Certification Test Prep Courses. Ex 11.7.3 Compute \(\lim_{n\to\infty} |a_n|^{1/n}\) for the series \(\sum 1/n^2\). The Alternating Series Test can be used only if the terms of the Which of the following is the 14th term of the sequence below? %PDF-1.5 /FirstChar 0 Estimating the Value of a Series In this section we will discuss how the Integral Test, Comparison Test, Alternating Series Test and the Ratio Test can, on occasion, be used to estimating the value of an infinite series. Sequences and Series: Comparison Test; Taylor Polynomials Practice; Power Series Practice; Calculus II Arc Length of Parametric Equations; 3 Dimensional Lines; Vectors Practice; Meanvariance SD - Mean Variance; Preview text. Calculus II-Sequences and Series. nn = 0. Chapter 10 : Series and Sequences. 62 0 obj 1 2, 1 4, 1 8, Sequences of values of this type is the topic of this rst section. /FontDescriptor 14 0 R 1000 1000 777.8 777.8 1000 1000 500 500 1000 1000 1000 777.8 1000 1000 611.1 611.1 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. SAT Practice Questions- All Maths; SAT Practice Test Questions- Reading , Writing and Language; KS 1-2 Math, Science and SAT . 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, \( \displaystyle \sum\limits_{n = 1}^\infty {\left( {n{2^n} - {3^{1 - n}}} \right)} \), \( \displaystyle \sum\limits_{n = 7}^\infty {\frac{{4 - n}}{{{n^2} + 1}}} \), \( \displaystyle \sum\limits_{n = 2}^\infty {\frac{{{{\left( { - 1} \right)}^{n - 3}}\left( {n + 2} \right)}}{{{5^{1 + 2n}}}}} \). Course summary; . /LastChar 127 Given that \( \displaystyle \sum\limits_{n = 0}^\infty {\frac{1}{{{n^3} + 1}}} = 1.6865\) determine the value of \( \displaystyle \sum\limits_{n = 2}^\infty {\frac{1}{{{n^3} + 1}}} \). Level up on all the skills in this unit and collect up to 2000 Mastery points! }\right\}_{n=0}^{\infty}\) converges or diverges. /FontDescriptor 17 0 R Most sections should have a range of difficulty levels in the problems although this will vary from section to section. 5.3.2 Use the integral test to determine the convergence of a series. << If L = 1, then the test is inconclusive. /Subtype/Type1 /Filter /FlateDecode Determine whether the sequence converges or diverges. endobj Which of the following sequences is NOT a geometric sequence? Infinite series are sums of an infinite number of terms. ]^e-V!2 F. /Type/Font Determine whether each series converges or diverges. What is the radius of convergence? This will work for a much wider variety of function than the method discussed in the previous section at the expense of some often unpleasant work. Ex 11.1.1 Compute \(\lim_{x\to\infty} x^{1/x}\). We will also give the Divergence Test for series in this section. Determine whether the following series converge or diverge. xWKoFWlojCpP NDED$(lq"g|3g6X_&F1BXIM5d gOwaN9c,r|9 207 0 obj <> endobj >> /FirstChar 0 /Type/Font in calculus coursesincluding Calculus, Calculus II, Calculus III, AP Calculus and Precalculus. Bottom line -- series are just a lot of numbers added together. 1111.1 472.2 555.6 1111.1 1511.1 1111.1 1511.1 1111.1 1511.1 1055.6 944.4 472.2 833.3 << 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 /BaseFont/UNJAYZ+CMR12 More on Sequences In this section we will continue examining sequences. Convergence/Divergence of Series In this section we will discuss in greater detail the convergence and divergence of infinite series. 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. 805.6 805.6 1277.8 1277.8 811.1 811.1 875 875 666.7 666.7 666.7 666.7 666.7 666.7 >> Harmonic series and p-series. /FontDescriptor 23 0 R Alternating series test. % For problems 1 - 3 perform an index shift so that the series starts at n = 3 n = 3. Applications of Series In this section we will take a quick look at a couple of applications of series. stream (answer), Ex 11.9.2 Find a power series representation for \(1/(1-x)^2\). Which rule represents the nth term in the sequence 9, 16, 23, 30? (answer), Ex 11.2.1 Explain why \(\sum_{n=1}^\infty {n^2\over 2n^2+1}\) diverges. 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 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! Comparison Test: This applies . Which one of these sequences is a finite sequence? 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. /Length 1247 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. 590.3 885.4 885.4 295.1 324.7 531.3 531.3 531.3 531.3 531.3 795.8 472.2 531.3 767.4 31 terms. (5 points) Evaluate the integral: Z 1 1 1 x2 dx = SOLUTION: The function 1/x2 is undened at x = 0, so we we must evaluate the im- proper integral as a limit. %PDF-1.5 % (answer), Ex 11.1.5 Determine whether \(\left\{{n+47\over\sqrt{n^2+3n}}\right\}_{n=1}^{\infty}\) converges or diverges. Solution. 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 % endobj << Each term is the product of the two previous terms. We also discuss differentiation and integration of power series. /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 Published by Wiley. We will also see how we can use the first few terms of a power series to approximate a function. In mathematical analysis, the alternating series test is the method used to show that an alternating series is convergent when its terms (1) decrease in absolute value, and (2) approach zero in the limit. Ex 11.1.2 Use the squeeze theorem to show that limn n! << /FontDescriptor 11 0 R << Ex 11.1.3 Determine whether {n + 47 n} . Ex 11.11.5 Show that \(e^x\) is equal to its Taylor series for all \(x\) by showing that the limit of the error term is zero as \(N\) approaches infinity. /LastChar 127 Learning Objectives. . How many bricks are in the 12th row? /Filter /FlateDecode Determine whether the series converge or diverge. n a n converges if and only if the integral 1 f ( x) d x converges. 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 2.(a). /Type/Font /FontDescriptor 20 0 R Root Test In this section we will discuss using the Root Test to determine if an infinite series converges absolutely or diverges. At this time, I do not offer pdfs for solutions to individual problems. endstream Integral Test In this section we will discuss using the Integral Test to determine if an infinite series converges or diverges. Which of the following represents the distance the ball bounces from the first to the seventh bounce with sigma notation? MULTIPLE CHOICE: Circle the best answer. (answer), Ex 11.10.9 Use a combination of Maclaurin series and algebraic manipulation to find a series centered at zero for \( x\cos (x^2)\). /FontDescriptor 8 0 R 1. %%EOF /Name/F5 MATH 126 Medians and Such. 5.3.1 Use the divergence test to determine whether a series converges or diverges. /LastChar 127 Legal. 531.3 531.3 531.3] 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! Ex 11.3.1 \(\sum_{n=1}^\infty {1\over n^{\pi/4}}\) (answer), Ex 11.3.2 \(\sum_{n=1}^\infty {n\over n^2+1}\) (answer), Ex 11.3.3 \(\sum_{n=1}^\infty {\ln n\over n^2}\) (answer), Ex 11.3.4 \(\sum_{n=1}^\infty {1\over n^2+1}\) (answer), Ex 11.3.5 \(\sum_{n=1}^\infty {1\over e^n}\) (answer), Ex 11.3.6 \(\sum_{n=1}^\infty {n\over e^n}\) (answer), Ex 11.3.7 \(\sum_{n=2}^\infty {1\over n\ln n}\) (answer), Ex 11.3.8 \(\sum_{n=2}^\infty {1\over n(\ln n)^2}\) (answer), Ex 11.3.9 Find an \(N\) so that \(\sum_{n=1}^\infty {1\over n^4}\) is between \(\sum_{n=1}^N {1\over n^4}\) and \(\sum_{n=1}^N {1\over n^4} + 0.005\). endobj 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. What if the interval is instead \([1,3/2]\)? Sequences and Numerical series. It turns out the answer is no. web manual for algebra 2 and pre calculus volume ii pre calculus for dummies jan 20 2021 oers an introduction to the principles of pre calculus covering such topics as functions law of sines and cosines identities sequences series and binomials algebra 2 homework practice workbook oct 29 2021 algebra ii practice tests varsity tutors - Nov 18 . xYKs6W(MCG:9iIO=(lkFRI$x$AMN/" J?~i~d cXf9o/r.&Lxy%/D-Yt+"LX]Sfp]Xl-aM_[6(*~mQbh*28AjZx0 =||. Quiz 2: 8 questions Practice what you've learned, and level up on the above skills. Ex 11.1.2 Use the squeeze theorem to show that \(\lim_{n\to\infty} {n!\over n^n}=0\). If the series is an alternating series, determine whether it converges absolutely, converges conditionally, or diverges. Choose your answer to the question and click 'Continue' to see how you did. Khan Academy is a 501(c)(3) nonprofit organization. 531.3 590.3 472.2 590.3 472.2 324.7 531.3 590.3 295.1 324.7 560.8 295.1 885.4 590.3 Ratio test. Ex 11.6.1 \(\sum_{n=1}^\infty (-1)^{n-1}{1\over 2n^2+3n+5}\) (answer), Ex 11.6.2 \(\sum_{n=1}^\infty (-1)^{n-1}{3n^2+4\over 2n^2+3n+5}\) (answer), Ex 11.6.3 \(\sum_{n=1}^\infty (-1)^{n-1}{\ln n\over n}\) (answer), Ex 11.6.4 \(\sum_{n=1}^\infty (-1)^{n-1} {\ln n\over n^3}\) (answer), Ex 11.6.5 \(\sum_{n=2}^\infty (-1)^n{1\over \ln n}\) (answer), Ex 11.6.6 \(\sum_{n=0}^\infty (-1)^{n} {3^n\over 2^n+5^n}\) (answer), Ex 11.6.7 \(\sum_{n=0}^\infty (-1)^{n} {3^n\over 2^n+3^n}\) (answer), Ex 11.6.8 \(\sum_{n=1}^\infty (-1)^{n-1} {\arctan n\over n}\) (answer). Then click 'Next Question' to answer the next question. A summary of all the various tests, as well as conditions that must be met to use them, we discussed in this chapter are also given in this section. 45 0 obj Ex 11.9.5 Find a power series representation for \(\int\ln(1-x)\,dx\). Strip out the first 3 terms from the series \( \displaystyle \sum\limits_{n = 1}^\infty {\frac{{{2^{ - n}}}}{{{n^2} + 1}}} \). nth-term test. 1000 500 500 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 The book contains eight practice tests five practice tests for Calculus AB and three practice tests for Calculus BC. stream 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. /Widths[606.7 816 748.3 679.6 728.7 811.3 765.8 571.2 652.8 598 757.6 622.8 552.8 Series The Basics In this section we will formally define an infinite series. 2 6 points 2. Given item A, which of the following would be the value of item B? 252 0 obj <>stream (b) Chapters include Linear When given a sum a[n], if the limit as n-->infinity does not exist or does not equal 0, the sum diverges. Therefore the radius of convergence is R = , and the interval of convergence is ( - , ). 9 0 obj The numbers used come from a sequence. A proof of the Root Test is also given. Given that n=0 1 n3 +1 = 1.6865 n = 0 1 n 3 + 1 = 1.6865 determine the value of n=2 1 n3 +1 . Contact us by phone at (877)266-4919, or by mail at 100ViewStreet#202, MountainView, CA94041. For each function, find the Maclaurin series or Taylor series centered at $a$, and the radius of convergence. We will also give many of the basic facts and properties well need as we work with sequences. It turns out the answer is no. Absolute Convergence In this section we will have a brief discussion on absolute convergence and conditionally convergent and how they relate to convergence of infinite series. Martha_Austin Teacher. copyright 2003-2023 Study.com. ,vEmO8/OuNVRaLPqB.*l. Math 129 - Calculus II. stream 666.7 1000 1000 1000 1000 1055.6 1055.6 1055.6 777.8 666.7 666.7 450 450 450 450 Integral Test: If a n = f ( n), where f ( x) is a non-negative non-increasing function, then. 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 L7s[AQmT*Z;HK%H0yqt1r8 To log in and use all the features of Khan Academy, please enable JavaScript in your browser. /Name/F3 endobj Determine whether each series converges absolutely, converges conditionally, or diverges. (answer), Ex 11.2.3 Explain why \(\sum_{n=1}^\infty {3\over n}\) diverges. In order to use either test the terms of the infinite series must be positive. /FirstChar 0 /Type/Font Here are a set of practice problems for the Series and Sequences chapter of the Calculus II notes. A ball is dropped from an unknown height (h) and it repeatedly bounces on the floor. /BaseFont/PSJLQR+CMEX10 /Type/Font 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 About this unit. The chapter headings refer to Calculus, Sixth Edition by Hughes-Hallett et al. 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 Use the Comparison Test to determine whether each series in exercises 1 - 13 converges or diverges. You appear to be on a device with a "narrow" screen width (, 2.4 Equations With More Than One Variable, 2.9 Equations Reducible to Quadratic in Form, 4.1 Lines, Circles and Piecewise Functions, 1.5 Trig Equations with Calculators, Part I, 1.6 Trig Equations with Calculators, Part II, 3.6 Derivatives of Exponential and Logarithm Functions, 3.7 Derivatives of Inverse Trig Functions, 4.10 L'Hospital's Rule and Indeterminate Forms, 5.3 Substitution Rule for Indefinite Integrals, 5.8 Substitution Rule for Definite Integrals, 6.3 Volumes of Solids of Revolution / Method of Rings, 6.4 Volumes of Solids of Revolution/Method of Cylinders, A.2 Proof of Various Derivative Properties, A.4 Proofs of Derivative Applications Facts, 7.9 Comparison Test for Improper Integrals, 9. 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 Ex 11.7.4 Compute \(\lim_{n\to\infty} |a_n|^{1/n}\) for the series \(\sum 1/n\). Calculus (single and multi-variable) Ordinary Differential equations (upto 2nd order linear with Laplace transforms, including Dirac Delta functions and Fourier Series. (answer), Ex 11.11.1 Find a polynomial approximation for \(\cos x\) on \([0,\pi]\), accurate to \( \pm 10^{-3}\) (answer), Ex 11.11.2 How many terms of the series for \(\ln x\) centered at 1 are required so that the guaranteed error on \([1/2,3/2]\) is at most \( 10^{-3}\)? 21 terms. Ex 11.7.9 Prove theorem 11.7.3, the root test. We will focus on the basic terminology, limits of sequences and convergence of sequences in this section. (answer), Ex 11.2.7 Compute \(\sum_{n=0}^\infty {3^{n+1}\over 7^{n+1}}\). Which of the following sequences is NOT a geometric sequence? Then we can say that the series diverges without having to do any extra work. (answer). If it converges, compute the limit. If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. Ex 11.8.1 \(\sum_{n=0}^\infty n x^n\) (answer), Ex 11.8.2 \(\sum_{n=0}^\infty {x^n\over n! 611.8 897.2 734 761.6 666.2 761.6 720.6 544 707.2 734 734 1006 734 734 598.4 272 Our mission is to provide a free, world-class education to anyone, anywhere. (answer). 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 x=S0 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