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. 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 %PDF-1.5 722.2 777.8 777.8 611.1 798.5 656.8 526.5 771.4 527.8 718.7 594.9 844.5 544.5 677.8 The Alternating Series Test can be used only if the terms of the series alternate in sign. Ex 11.7.2 Compute \(\lim_{n\to\infty} |a_{n+1}/a_n|\) for the series \(\sum 1/n\). PDF Schaums Outline Of Differential Equations 4th Edition Schaums Outline xTn0+,ITi](N@ fH2}W"UG'.% Z#>y{!9kJ+ Ex 11.11.4 Show that \(\cos x\) is equal to its Taylor series for all \(x\) by showing that the limit of the error term is zero as N approaches infinity. >> /Name/F5 MATH 126 Medians and Such. At this time, I do not offer pdf's for . At this time, I do not offer pdf's for solutions to individual problems. /BaseFont/UNJAYZ+CMR12 Then click 'Next Question' to answer the next question. Math Journey: Calculus, ODEs, Linear Algebra and Beyond MULTIPLE CHOICE: Circle the best answer. n = 1 n2 + 2n n3 + 3n2 + 1. /BaseFont/CQGOFL+CMSY10 Ex 11.1.2 Use the squeeze theorem to show that limn n! In the previous section, we determined the convergence or divergence of several series by . Math 129 - Calculus II. xWKoFWlojCpP NDED$(lq"g|3g6X_&F1BXIM5d gOwaN9c,r|9 /Length 1247 Chapter 10 : Series and Sequences. Determine whether the series is convergent or divergent. We also discuss differentiation and integration of power series. Series The Basics In this section we will formally define an infinite series. /Name/F2 If it converges, compute the limit. Khan Academy is a 501(c)(3) nonprofit organization. /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 Which of the following sequences is NOT a geometric sequence? /FontDescriptor 8 0 R 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. (answer), Ex 11.1.5 Determine whether \(\left\{{n+47\over\sqrt{n^2+3n}}\right\}_{n=1}^{\infty}\) converges or diverges. Which one of these sequences is a finite sequence? /Filter /FlateDecode Ex 11.7.9 Prove theorem 11.7.3, the root test. 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 All rights reserved. Then determine if the series converges or diverges. The following is a list of worksheets and other materials related to Math 129 at the UA. Legal. 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. 12 0 obj 26 0 obj 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 Calculus II - Series - The Basics (Practice Problems) - Lamar University 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. 531.3 531.3 531.3] The Alternating Series Test can be used only if the terms of the /Subtype/Type1 Use the Comparison Test to determine whether each series in exercises 1 - 13 converges or diverges. PDF Practice Problems Series & Sequences - MR. SOLIS' WEEBLY Our mission is to provide a free, world-class education to anyone, anywhere. We will illustrate how partial sums are used to determine if an infinite series converges or diverges. (answer), Ex 11.2.5 Compute \(\sum_{n=0}^\infty {3\over 2^n}+ {4\over 5^n}\). (answer), Ex 11.2.3 Explain why \(\sum_{n=1}^\infty {3\over n}\) diverges. A proof of the Ratio Test is also given. 1. L7s[AQmT*Z;HK%H0yqt1r8 /Filter /FlateDecode Good luck! >> 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 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 raVQ1CKD3`
rO:H\hL[+?zWl'oDpP% bhR5f7RN
`1= SJt{p9kp5,W+Y.e7) Zy\BP>+``;qI^%$x=%f0+!.=Q7HgbjfCVws,NL)%"pcS^ {tY}vf~T{oFe{nB\bItw$nku#pehXWn8;ZW]/v_nF787nl{ y/@U581$&DN>+gt Infinite sequences and series | AP/College Calculus BC - Khan Academy 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. /Length 200 (answer). If you're seeing this message, it means we're having trouble loading external resources on our website. If L = 1, then the test is inconclusive. Ratio Test In this section we will discuss using the Ratio Test to determine if an infinite series converges absolutely or diverges. 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. (answer), Ex 11.2.6 Compute \(\sum_{n=0}^\infty {4^{n+1}\over 5^n}\). 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 endobj 1) \(\displaystyle \sum^_{n=1}a_n\) where \(a_n=\dfrac{2}{n . in calculus coursesincluding Calculus, Calculus II, Calculus III, AP Calculus and Precalculus. endobj Solution. !A1axw)}p]WgxmkFftu When you have completed the free practice test, click 'View Results' to see your results. 45 0 obj Parametric equations, polar coordinates, and vector-valued functions Calculator-active practice: Parametric equations, polar coordinates, . Calculus II - Series & Sequences (Practice Problems) - Lamar University copyright 2003-2023 Study.com. copyright 2003-2023 Study.com. 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). Comparison tests. 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 { "11.01:_Prelude_to_Sequences_and_Series" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "11.02:_Sequences" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "11.03:_Series" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "11.04:_The_Integral_Test" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "11.05:_Alternating_Series" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "11.06:_Comparison_Test" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "11.07:_Absolute_Convergence" : "property get [Map 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endobj Note that some sections will have more problems than others and some will have more or less of a variety of problems. Ex 11.10.8 Find the first four terms of the Maclaurin series for \(\tan x\) (up to and including the \( x^3\) term). 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 . /Length 465 PDF Read Free Answers To Algebra 2 Practice B Ellipses /BaseFont/PSJLQR+CMEX10 Contact us by phone at (877)266-4919, or by mail at 100ViewStreet#202, MountainView, CA94041. 5.3.1 Use the divergence test to determine whether a series converges or diverges. Which of the following is the 14th term of the sequence below? (answer), Ex 11.9.3 Find a power series representation for \( 2/(1-x)^3\). If the series is an alternating series, determine whether it converges absolutely, converges conditionally, or diverges. Which of the following sequences follows this formula. Contact us by phone at (877)266-4919, or by mail at 100ViewStreet#202, MountainView, CA94041. Ex 11.1.2 Use the squeeze theorem to show that \(\lim_{n\to\infty} {n!\over n^n}=0\). (answer), Ex 11.11.3 Find the first three nonzero terms in the Taylor series for \(\tan x\) on \([-\pi/4,\pi/4]\), and compute the guaranteed error term as given by Taylor's theorem. 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. (answer). /Filter /FlateDecode Therefore the radius of convergence is R = , and the interval of convergence is ( - , ). 326.4 272 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 272 272 883.8 992.6 761.6 272 272 489.6 816 489.6 816 761.6 272 380.8 380.8 489.6 761.6 272 We will determine if a sequence in an increasing sequence or a decreasing sequence and hence if it is a monotonic sequence. 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. Each term is the difference of the previous two terms. 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? (answer), Ex 11.2.8 Compute \(\sum_{n=1}^\infty \left({3\over 5}\right)^n\). 207 0 obj
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555.6 577.8 577.8 597.2 597.2 736.1 736.1 527.8 527.8 583.3 583.3 583.3 583.3 750 Then click 'Next Question' to answer the . Published by Wiley. 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. 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 The sum of the steps forms an innite series, the topic of Section 10.2 and the rest of Chapter 10. Psychological Research & Experimental Design, All Teacher Certification Test Prep Courses. 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 Level up on all the skills in this unit and collect up to 2000 Mastery points! 11.E: Sequences and Series (Exercises) These are homework exercises to accompany David Guichard's "General Calculus" Textmap. . Derivatives, Integrals, Sequences & Series, and Vector Valued Functions. Calculus II For Dummies Cheat Sheet - dummies /Filter /FlateDecode Chapters include Linear 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 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. 4 avwo/MpLv)
_C>5p*)i=^m7eE. Determine whether the following series converge or diverge. Here are a set of practice problems for the Series and Sequences chapter of the Calculus II notes. Given that \( \displaystyle \sum\limits_{n = 0}^\infty {\frac{1}{{{n^3} + 1}}} = 1.6865\) determine the value of \( \displaystyle \sum\limits_{n = 2}^\infty {\frac{1}{{{n^3} + 1}}} \). endstream Question 5 5. Solving My Calc 2 Exam#3 (Sequence, Infinite Series & Power Series) hbbd```b``~"A$"
"Y`L6`RL,-`sA$w64= f[" RLMu;@jAl[`3H^Ne`?$4 Calculus II - Sequences and Series Flashcards | Quizlet 666.7 1000 1000 1000 1000 1055.6 1055.6 1055.6 777.8 666.7 666.7 450 450 450 450 Learn how this is possible, how we can tell whether a series converges, and how we can explore convergence in Taylor and Maclaurin series. 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. 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 Integral test. (answer). /FirstChar 0 826.4 531.3 958.7 1076.8 826.4 295.1 295.1 531.3 885.4 531.3 885.4 826.4 295.1 413.2 (1 point) Is the integral Z 1 1 1 x2 dx an improper integral? Calculus 2. 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 /BaseFont/BPHBTR+CMMI12 We will also determine a sequence is bounded below, bounded above and/or bounded. 62 0 obj /Subtype/Type1 Calculus (single and multi-variable) Ordinary Differential equations (upto 2nd order linear with Laplace transforms, including Dirac Delta functions and Fourier Series. PDF M 172 - Calculus II - Chapter 10 Sequences and Series We also derive some well known formulas for Taylor series of \({\bf e}^{x}\) , \(\cos(x)\) and \(\sin(x)\) around \(x=0\). /Subtype/Type1 Alternating series test - Wikipedia Research Methods Midterm. 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 PDF Review Sheet for Calculus 2 Sequences and Series - Derrick Chung You may also use any of these materials for practice. AP is a registered trademark of the College Board, which has not reviewed this resource. 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 More on Sequences In this section we will continue examining sequences. )^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 \(\ln 2\). Our mission is to provide a free, world-class education to anyone, anywhere. Math 106 (Calculus II): old exams. OR sequences are lists of numbers, where the numbers may or may not be determined by a pattern. 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 /FirstChar 0 272 816 544 489.6 544 516.8 380.8 386.2 380.8 544 516.8 707.2 516.8 516.8 435.2 489.6 Convergence/Divergence of Series In this section we will discuss in greater detail the convergence and divergence of infinite series. Don't all infinite series grow to infinity? Sequences & Series in Calculus Chapter Exam - Study.com 6.5E: Exercises for Comparison Test - Mathematics LibreTexts Harmonic series and p-series. >> (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 \(\sum_{n=2}^N {1\over n(\ln n)^2} + 0.005\). 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. 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. SAT Practice Questions- All Maths; SAT Practice Test Questions- Reading , Writing and Language; KS 1-2 Math, Science and SAT . Complementary General calculus exercises can be found for other Textmaps and can be accessed here. Defining convergent and divergent infinite series, Determining absolute or conditional convergence, Finding Taylor polynomial approximations of functions, Radius and interval of convergence of power series, Finding Taylor or Maclaurin series for a function. A review of all series tests. Which is the finite sequence of four multiples of 9, starting with 9? /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 A proof of the Root Test is also given. /Name/F1 )Ltgx?^eaT'&+n+hN4*D^UR!8UY@>LqS%@Cp/-12##DR}miBw6"ja+WjU${IH$5j!j-I1 We will also see how we can use the first few terms of a power series to approximate a function. Choosing a Convergence Test | Calculus II - Lumen Learning stream Part II. << UcTIjeB#vog-TM'FaTzG(:k-BNQmbj}'?^h<=XgS/]o4Ilv%Jm >> endstream PDF Arithmetic Sequences And Series Practice Problems %PDF-1.5
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Ex 11.1.1 Compute \(\lim_{x\to\infty} x^{1/x}\). /Subtype/Type1 %PDF-1.2 (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 \( 10^{-3}\)? Choose your answer to the question and click 'Continue' to see how you did. /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 /LastChar 127 Here is a list of all the sections for which practice problems have been written as well as a brief description of the material covered in the notes for that particular section. When you have completed the free practice test, click 'View Results' to see your results. ]^e-V!2
F. Example 1. Comparison Test: This applies . 611.8 897.2 734 761.6 666.2 761.6 720.6 544 707.2 734 734 1006 734 734 598.4 272 Integral Test In this section we will discuss using the Integral Test to determine if an infinite series converges or diverges. Sequences & Series in Calculus Chapter Exam. 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. Strip out the first 3 terms from the series n=1 2n n2 +1 n = 1 2 n n 2 + 1. Complementary General calculus exercises can be found for other Textmaps and can be accessed here. /FirstChar 0 556.5 425.2 527.8 579.5 613.4 636.6 609.7 458.2 577.1 808.9 505 354.2 641.4 979.2 We will focus on the basic terminology, limits of sequences and convergence of sequences in this section. Worksheets. 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. Ratio test. >> endstream Good luck! 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). bmkraft7. Choose your answer to the question and click 'Continue' to see how you did. 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 \(\sum_{n=1}^N {1\over n^4} + 0.005\). Absolute and conditional convergence. /FirstChar 0 AP Calculus AB and BC: Chapter 9 -Infinite Sequences and Series : 9.4 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. 441.3 461.2 353.6 557.3 473.4 699.9 556.4 477.4 454.9 312.5 377.9 623.4 489.6 272] Your instructor might use some of these in class. << endobj Calculus II - Series & Sequences (Practice Problems) - Lamar University
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