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I have not learned series solutions nor special functions which I see is the next step in this chapter) Linear Algebra (self-taught from Hoffman and Kunze. 238 0 obj <>/Filter/FlateDecode/ID[<09CA7BCBAA751546BDEE3FEF56AF7BFA>]/Index[207 46]/Info 206 0 R/Length 137/Prev 582846/Root 208 0 R/Size 253/Type/XRef/W[1 3 1]>>stream (answer), Ex 11.2.9 Compute \(\sum_{n=1}^\infty {3^n\over 5^{n+1}}\). >> The practice tests are composed xWKoFWlojCpP NDED$(lq"g|3g6X_&F1BXIM5d gOwaN9c,r|9 . 21 terms. If the series is an alternating series, determine whether it converges absolutely, converges conditionally, or diverges. 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 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 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 \( 2/(1-x)^3\). Each term is the difference of the previous two terms. 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. 1. Calculus II-Sequences and Series. endobj 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 . xYKs6W(MCG:9iIO=(lkFRI$x$AMN/" J?~i~d cXf9o/r.&Lxy%/D-Yt+"LX]Sfp]Xl-aM_[6(*~mQbh*28AjZx0 =||. << 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 Calculus II Series - Things to Consider - California State University If it con-verges, nd the limit. 8 0 obj Images. 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 24 0 obj Determine whether the following series converge or diverge. Your instructor might use some of these in class. About this unit. 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 /Subtype/Type1 n = 1 n 2 + 2 n n 3 + 3 n . n = 1 n2 + 2n n3 + 3n2 + 1. /BaseFont/VMQJJE+CMR8 We will examine Geometric Series, Telescoping Series, and Harmonic Series. Ex 11.7.1 Compute \(\lim_{n\to\infty} |a_{n+1}/a_n|\) for the series \(\sum 1/n^2\). 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 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. If youd like a pdf document containing the solutions the download tab above contains links to pdfs containing the solutions for the full book, chapter and section. 777.8 444.4 444.4 444.4 611.1 777.8 777.8 777.8 777.8] If you're seeing this message, it means we're having trouble loading external resources on our website. (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 \(\sum_{n=0}^N {1\over e^n} + 10^{-4}\). What is the 83rd term of the sequence 91, 87, 83, 79, ( = a. If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. /Length 2492 Sequences can be thought of as functions whose domain is the set of integers. /FirstChar 0 PDF FINAL EXAM CALCULUS 2 - Department of Mathematics (a) $\sum_{n=1}^{\infty} \frac{(-1)^n}{\sqrt{n}}$ (b) $\sum_{n=1}^{\infty}(-1)^n \frac{n}{2 n-1}$ MULTIPLE CHOICE: Circle the best answer. 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. Series Infinite geometric series: Series nth-term test: Series Integral test: Series Harmonic series and p-series: Series Comparison tests: . /BaseFont/UNJAYZ+CMR12 /LastChar 127 PDF Review Sheet for Calculus 2 Sequences and Series - Derrick Chung 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. Which of the following sequences follows this formula. If it converges, compute the limit. /BaseFont/BPHBTR+CMMI12 PDF M 172 - Calculus II - Chapter 10 Sequences and Series Which rule represents the nth term in the sequence 9, 16, 23, 30? ,vEmO8/OuNVRaLPqB.*l. (answer), Ex 11.1.5 Determine whether \(\left\{{n+47\over\sqrt{n^2+3n}}\right\}_{n=1}^{\infty}\) converges or diverges. (answer). /Length 200 /Type/Font Then determine if the series converges or diverges. The Alternating Series Test can be used only if the terms of the All rights reserved. Worksheets. /FontDescriptor 23 0 R Good luck! 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! (answer), Ex 11.10.9 Use a combination of Maclaurin series and algebraic manipulation to find a series centered at zero for \( x\cos (x^2)\). 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. 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. Parametric equations, polar coordinates, and vector-valued functions Calculator-active practice: Parametric equations, polar coordinates, . Sequences & Series in Calculus Chapter Exam - Study.com Taylor Series In this section we will discuss how to find the Taylor/Maclaurin Series for a function. 21 0 obj Bottom line -- series are just a lot of numbers added together. If you're seeing this message, it means we're having trouble loading external resources on our website. /Filter /FlateDecode /Subtype/Type1 % Binomial Series In this section we will give the Binomial Theorem and illustrate how it can be used to quickly expand terms in the form \( \left(a+b\right)^{n}\) when \(n\) is an integer. endobj Remark. Math Journey: Calculus, ODEs, Linear Algebra and Beyond We will illustrate how we can find a series representation for indefinite integrals that cannot be evaluated by any other method. 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, \( \displaystyle \sum\limits_{n = 1}^\infty {\left( {n{2^n} - {3^{1 - n}}} \right)} \), \( \displaystyle \sum\limits_{n = 7}^\infty {\frac{{4 - n}}{{{n^2} + 1}}} \), \( \displaystyle \sum\limits_{n = 2}^\infty {\frac{{{{\left( { - 1} \right)}^{n - 3}}\left( {n + 2} \right)}}{{{5^{1 + 2n}}}}} \). %PDF-1.2 Series are sums of multiple terms. The Integral Test can be used on a infinite series provided the terms of the series are positive and decreasing. hbbd```b``~"A$" "Y`L6`RL,-`sA$w64= f[" RLMu;@jAl[`3H^Ne`?$4 5.3.2 Use the integral test to determine the convergence of a series. Math 106 (Calculus II): old exams. Calculus II - Sequences and Series Flashcards | Quizlet Example 1. 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 Determine whether the series is convergent or divergent. Maclaurin series of e, sin(x), and cos(x). Ex 11.7.2 Compute \(\lim_{n\to\infty} |a_{n+1}/a_n|\) for the series \(\sum 1/n\). 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 OR sequences are lists of numbers, where the numbers may or may not be determined by a pattern. Choose the equation below that represents the rule for the nth term of the following geometric sequence: 128, 64, 32, 16, 8, . )^2\over n^n}\) (answer). (answer). (answer). 12 0 obj 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. Harmonic series and p-series. To log in and use all the features of Khan Academy, please enable JavaScript in your browser. nn = 0. Sequences and Series for Calculus Chapter Exam - Study.com x[[o6~cX/e`ElRm'1%J$%v)tb]1U2sRV}.l%s\Y UD+q}O+J 6.5E: Exercises for Comparison Test - Mathematics LibreTexts /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 >> Ex 11.9.5 Find a power series representation for \(\int\ln(1-x)\,dx\). Math 1242: Calculus II - University of North Carolina at Charlotte endstream We will also give many of the basic facts and properties well need as we work with sequences. 1000 1000 777.8 777.8 1000 1000 500 500 1000 1000 1000 777.8 1000 1000 611.1 611.1 /Name/F6 /Filter /FlateDecode 777.8 777.8] 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 The Alternating Series Test can be used only if the terms of the series alternate in sign. Ex 11.1.3 Determine whether \(\{\sqrt{n+47}-\sqrt{n}\}_{n=0}^{\infty}\) converges or diverges. Determine whether the series converge or diverge. The LibreTexts libraries arePowered by NICE CXone Expertand are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. 26 0 obj Here are a set of practice problems for the Series and Sequences chapter of the Calculus II notes. >> 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 /Filter /FlateDecode When you have completed the free practice test, click 'View Results' to see your results. /Name/F5 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). (answer). Strategies for Testing Series - University of Texas at Austin Donate or volunteer today! /LastChar 127 In other words, a series is the sum of a sequence. Calc II: Practice Final Exam 5 and our series converges because P nbn is a p-series with p= 4=3 >1: (b) X1 n=1 lnn n3 Set f(x) = lnx x3 and check that f0= 43x lnx+ x 4 <0 Series | Calculus 2 | Math | Khan Academy At this time, I do not offer pdfs for solutions to individual problems. 4 avwo/MpLv) _C>5p*)i=^m7eE. 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. Alternating series test - Wikipedia PDF Schaums Outline Of Differential Equations 4th Edition Schaums Outline Then click 'Next Question' to answer the next question. (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 \( 10^{-3}\)? /Subtype/Type1 Complementary General calculus exercises can be found for other Textmaps and can be accessed here. Don't all infinite series grow to infinity? 489.6 489.6 272 272 761.6 489.6 761.6 489.6 516.9 734 743.9 700.5 813 724.8 633.9 We will also give many of the basic facts, properties and ways we can use to manipulate a series. 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 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. >> %%EOF Choosing a Convergence Test | Calculus II - Lumen Learning >> A proof of the Alternating Series Test is also given. Which of the following is the 14th term of the sequence below? Published by Wiley. AP is a registered trademark of the College Board, which has not reviewed this resource. (answer), Ex 11.2.6 Compute \(\sum_{n=0}^\infty {4^{n+1}\over 5^n}\). << (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. We will also give the Divergence Test for series in this section. 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 When you have completed the free practice test, click 'View Results' to see your results. Martha_Austin Teacher. /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 Ex 11.7.4 Compute \(\lim_{n\to\infty} |a_n|^{1/n}\) for the series \(\sum 1/n\). !A1axw)}p]WgxmkFftu /Name/F1 /Name/F2 Some infinite series converge to a finite value. It turns out the answer is no. 531.3 590.3 472.2 590.3 472.2 324.7 531.3 590.3 295.1 324.7 560.8 295.1 885.4 590.3 272 816 544 489.6 544 516.8 380.8 386.2 380.8 544 516.8 707.2 516.8 516.8 435.2 489.6 Chapters include Linear Accessibility StatementFor more information contact us atinfo@libretexts.org. Solving My Calc 2 Exam#3 (Sequence, Infinite Series & Power Series) If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. /Type/Font The numbers used come from a sequence. Absolute and conditional convergence. 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 (answer), Ex 11.4.6 Approximate \(\sum_{n=1}^\infty (-1)^{n-1}{1\over n^4}\) to two decimal places. 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 \(\sum_{n=1}^N {1\over n^4} + 0.005\). stream YesNo 2.(b). This page titled 11.E: Sequences and Series (Exercises) is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by David Guichard. |: The Ratio Test shows us that regardless of the choice of x, the series converges. A brick wall has 60 bricks in the first row, but each row has 3 fewer bricks than the previous one. Comparison tests. 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. PDF Calc II: Practice Final Exam - Columbia University Quiz 1: 5 questions Practice what you've learned, and level up on the above skills. 9.8 Power Series Chapter 9 Sequences and Series Calculus II 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. /FirstChar 0 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). 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. 31 terms. Let the factor without dx equal u and the factor with dx equal dv. (answer), Ex 11.1.4 Determine whether \(\left\{{n^2+1\over (n+1)^2}\right\}_{n=0}^{\infty}\) converges or diverges. Sequences review (practice) | Series | Khan Academy Quiz 2: 8 questions Practice what you've learned, and level up on the above skills. We will determine if a sequence in an increasing sequence or a decreasing sequence and hence if it is a monotonic sequence. 5.3.3 Estimate the value of a series by finding bounds on its remainder term. 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. For each function, find the Maclaurin series or Taylor series centered at $a$, and the radius of convergence. All other trademarks and copyrights are the property of their respective owners. We will also determine a sequence is bounded below, bounded above and/or bounded. << 18 0 obj These are homework exercises to accompany David Guichard's "General Calculus" Textmap.