Ratio Test In this section we will discuss using the Ratio Test to determine if an infinite series converges absolutely or diverges. /LastChar 127 /FirstChar 0 Complementary General calculus exercises can be found for other Textmaps and can be accessed here. %PDF-1.5 % /BaseFont/SFGTRF+CMSL12 The following is a list of worksheets and other materials related to Math 129 at the UA. Ex 11.7.9 Prove theorem 11.7.3, the root test. /FirstChar 0 9 0 obj Learn for free about math, art, computer programming, economics, physics, chemistry, biology, medicine, finance, history, and more. Remark. nth-term test. After each bounce, the ball reaches a height that is 2/3 of the height from which it previously fell. We will focus on the basic terminology, limits of sequences and convergence of sequences in this section. When you have completed the free practice test, click 'View Results' to see your results. Find the radius and interval of convergence for each series. You may also use any of these materials for practice. >> (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 \(\sum_{n=0}^N {1\over e^n} + 10^{-4}\). (answer). Comparison tests. If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. PDF FINAL EXAM CALCULUS 2 - Department of Mathematics 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 endobj (answer), Ex 11.9.2 Find a power series representation for \(1/(1-x)^2\). 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. 45 0 obj Images. 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 /BaseFont/CQGOFL+CMSY10 Ratio test. S.QBt'(d|/"XH4!qbnEriHX)Gs2qN/G jq8$$< Note that some sections will have more problems than others and some will have more or less of a variety of problems. 5.3.2 Use the integral test to determine the convergence of a series. 18 0 obj Which of the following is the 14th term of the sequence below? 668.3 724.7 666.7 666.7 666.7 666.7 666.7 611.1 611.1 444.4 444.4 444.4 444.4 500 n a n converges if and only if the integral 1 f ( x) d x converges. It turns out the answer is no. Chapter 10 : Series and Sequences. Free Practice Test Instructions: Choose your answer to the question and click 'Continue' to see how you did. Calculus II - Series & Sequences (Practice Problems) - Lamar University copyright 2003-2023 Study.com. Comparison Test: This applies . When you have completed the free practice test, click 'View Results' to see your results. >> %PDF-1.2 Good luck! If L = 1, then the test is inconclusive. The sum of the steps forms an innite series, the topic of Section 10.2 and the rest of Chapter 10. Convergence/Divergence of Series In this section we will discuss in greater detail the convergence and divergence of 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 2 6 points 2. Applications of Series In this section we will take a quick look at a couple of applications of series. At this time, I do not offer pdf's for . 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 Parametric Equations and Polar Coordinates, 9.5 Surface Area with Parametric Equations, 9.11 Arc Length and Surface Area Revisited, 10.7 Comparison Test/Limit Comparison Test, 12.8 Tangent, Normal and Binormal Vectors, 13.3 Interpretations of Partial Derivatives, 14.1 Tangent Planes and Linear Approximations, 14.2 Gradient Vector, Tangent Planes and Normal Lines, 15.3 Double Integrals over General Regions, 15.4 Double Integrals in Polar Coordinates, 15.6 Triple Integrals in Cylindrical Coordinates, 15.7 Triple Integrals in Spherical Coordinates, 16.5 Fundamental Theorem for Line Integrals, 3.8 Nonhomogeneous Differential Equations, 4.5 Solving IVP's with Laplace Transforms, 7.2 Linear Homogeneous Differential Equations, 8. All other trademarks and copyrights are the property of their respective owners. For problems 1 - 3 perform an index shift so that the series starts at n = 3 n = 3. 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 1 2, 1 4, 1 8, Sequences of values of this type is the topic of this rst section. Here are a set of practice problems for the Series and Sequences chapter of the Calculus II notes. 21 0 obj The numbers used come from a sequence. If it converges, compute the limit. (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 \( 10^{-3}\)? /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 Determine whether each series converges absolutely, converges conditionally, or diverges. /Subtype/Type1 Ex 11.7.2 Compute \(\lim_{n\to\infty} |a_{n+1}/a_n|\) for the series \(\sum 1/n\). Calculus II - Sequences and Series Flashcards | Quizlet MULTIPLE CHOICE: Circle the best answer. << Good luck! My calculus 2 exam on sequence, infinite series & power seriesThe exam: https://bit.ly/36OHYcsAll the convergence tests: https://bit.ly/2IzqokhBest friend an. Series Infinite geometric series: Series nth-term test: Series Integral test: Series Harmonic series and p-series: Series Comparison tests: . 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 Sequences review (practice) | Series | Khan Academy 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 Then we can say that the series diverges without having to do any extra work. 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. 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, \( \displaystyle \sum\limits_{n = 1}^\infty {\left( {n{2^n} - {3^{1 - n}}} \right)} \), \( \displaystyle \sum\limits_{n = 7}^\infty {\frac{{4 - n}}{{{n^2} + 1}}} \), \( \displaystyle \sum\limits_{n = 2}^\infty {\frac{{{{\left( { - 1} \right)}^{n - 3}}\left( {n + 2} \right)}}{{{5^{1 + 2n}}}}} \). /LastChar 127 Contact us by phone at (877)266-4919, or by mail at 100ViewStreet#202, MountainView, CA94041. If you . % 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. . 531.3 531.3 531.3] ,vEmO8/OuNVRaLPqB.*l. Integral Test: If a n = f ( n), where f ( x) is a non-negative non-increasing function, then. endstream endobj startxref /FontDescriptor 23 0 R (answer), Ex 11.2.9 Compute \(\sum_{n=1}^\infty {3^n\over 5^{n+1}}\). %|S#?\A@D-oS)lW=??nn}y]Tb!!o_=;]ha,J[. Maclaurin series of e, sin(x), and cos(x). A proof of the Alternating Series Test is also given. The practice tests are composed When you have completed the free practice test, click 'View Results' to see your results. How many bricks are in the 12th row? 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 . Worksheets. xYKs6W(MCG:9iIO=(lkFRI$x$AMN/" J?~i~d cXf9o/r.&Lxy%/D-Yt+"LX]Sfp]Xl-aM_[6(*~mQbh*28AjZx0 =||. Taylor Series In this section we will discuss how to find the Taylor/Maclaurin Series for a function. PDF Practice Problems Series & Sequences - MR. SOLIS' WEEBLY Solution. 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@ " PDF Review Sheet for Calculus 2 Sequences and Series - Derrick Chung AP Calculus AB and BC: Chapter 9 -Infinite Sequences and Series : 9.2 << We also discuss differentiation and integration of power series. Let the factor without dx equal u and the factor with dx equal dv. Then click 'Next Question' to answer the next question. PDF Read Free Answers To Algebra 2 Practice B Ellipses Don't all infinite series grow to infinity? /Name/F6 (answer), Ex 11.2.7 Compute \(\sum_{n=0}^\infty {3^{n+1}\over 7^{n+1}}\). Proofs for both tests are also given. Mediansandsuch - Medians - MATH 126 Medians and Such Let X be - Studocu If the series is an alternating series, determine whether it converges absolutely, converges conditionally, or diverges. /LastChar 127 Then click 'Next Question' to answer the . (1 point) Is the integral Z 1 1 1 x2 dx an improper integral? 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 /Type/Font 252 0 obj <>stream Ex 11.7.4 Compute \(\lim_{n\to\infty} |a_n|^{1/n}\) for the series \(\sum 1/n\). At this time, I do not offer pdf's for solutions to individual problems. The Alternating Series Test can be used only if the terms of the Premium members get access to this practice exam along with our entire library of lessons taught by subject matter experts. /Length 465 It turns out the answer is no. 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 We will illustrate how we can find a series representation for indefinite integrals that cannot be evaluated by any other method. 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. Premium members get access to this practice exam along with our entire library of lessons taught by subject matter experts. 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 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. /Name/F1 If you're seeing this message, it means we're having trouble loading external resources on our website. /LastChar 127 >> Published by Wiley. In exercises 3 and 4, do not attempt to determine whether the endpoints are in the interval of convergence. Quiz 2: 8 questions Practice what you've learned, and level up on the above skills. Good luck! /FontDescriptor 20 0 R (answer). \ _* %l~G"tytO(J*l+X@ uE: m/ ~&Q24Nss(7F!ky=4 Mijo8t;v