Which one of these sequences is a finite sequence? 5.3.1 Use the divergence test to determine whether a series converges or diverges. 5.3.3 Estimate the value of a series by finding bounds on its remainder term. Calculus 2. 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. 68 0 obj << 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. 750 750 750 1044.4 1044.4 791.7 791.7 583.3 583.3 638.9 638.9 638.9 638.9 805.6 805.6 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. copyright 2003-2023 Study.com. 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 Complementary General calculus exercises can be found for other Textmaps and can be accessed here. /Filter[/FlateDecode] Root Test In this section we will discuss using the Root Test to determine if an infinite series converges absolutely or diverges. To log in and use all the features of Khan Academy, please enable JavaScript in your browser. If a geometric series begins with the following term, what would the next term be? (answer), Ex 11.2.8 Compute \(\sum_{n=1}^\infty \left({3\over 5}\right)^n\). Harmonic series and p-series. (answer), Ex 11.9.4 Find a power series representation for \( 1/(1-x)^3\). Then click 'Next Question' to answer the next question. 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. /Filter /FlateDecode /Name/F3 What is the sum of all the even integers from 2 to 250? )^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 \(\ln 2\). 8 0 obj 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 The Alternating Series Test can be used only if the terms of the series alternate in sign. 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 %PDF-1.5 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. endobj Khan Academy is a 501(c)(3) nonprofit organization. /Length 2492 endobj (answer), Ex 11.10.10 Use a combination of Maclaurin series and algebraic manipulation to find a series centered at zero for \( xe^{-x}\). /Length 200 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. Use the Comparison Test to determine whether each series in exercises 1 - 13 converges or diverges. << Then click 'Next Question' to answer the next question. Khan Academy is a nonprofit with the mission of providing a free, world-class education for anyone, anywhere. 1000 1000 777.8 777.8 1000 1000 500 500 1000 1000 1000 777.8 1000 1000 611.1 611.1 stream Which of the following is the 14th term of the sequence below? All other trademarks and copyrights are the property of their respective owners. Contact us by phone at (877)266-4919, or by mail at 100ViewStreet#202, MountainView, CA94041. 722.6 693.1 833.5 795.8 382.6 545.5 825.4 663.6 972.9 795.8 826.4 722.6 826.4 781.6 508.8 453.8 482.6 468.9 563.7 334 405.1 509.3 291.7 856.5 584.5 470.7 491.4 434.1 If L = 1, then the test is inconclusive. 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 n = 1 n2 + 2n n3 + 3n2 + 1. Determine whether each series converges absolutely, converges conditionally, or diverges. Most sections should have a range of difficulty levels in the problems although this will vary from section to section. When you have completed the free practice test, click 'View Results' to see your results. /BaseFont/SFGTRF+CMSL12 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? 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. Part II. 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). /Length 1247 endobj 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 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. { "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 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MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()" }, [ "article:topic", "authorname:guichard", "license:ccbyncsa", "showtoc:no" ], https://math.libretexts.org/@app/auth/3/login?returnto=https%3A%2F%2Fmath.libretexts.org%2FBookshelves%2FCalculus%2FBook%253A_Calculus_(Guichard)%2F11%253A_Sequences_and_Series%2F11.E%253A_Sequences_and_Series_(Exercises), \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}}}\) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\). >> The numbers used come from a sequence. If you're seeing this message, it means we're having trouble loading external resources on our website. 1 2 + 1 4 + 1 8 + = n=1 1 2n = 1 We will need to be careful, but it turns out that we can . nth-term test. If it converges, compute the limit. hbbd```b``~"A$" "Y`L6`RL,-`sA$w64= f[" RLMu;@jAl[`3H^Ne`?$4 /Type/Font Math 129 - Calculus II. Bottom line -- series are just a lot of numbers added together. In exercises 3 and 4, do not attempt to determine whether the endpoints are in the interval of convergence. /LastChar 127 >> /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 2 6 points 2. /BaseFont/CQGOFL+CMSY10 Ex 11.1.2 Use the squeeze theorem to show that limn n! 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! 777.8 777.8] 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 Each term is the product of the two previous terms. /Widths[611.8 816 761.6 679.6 652.8 734 707.2 761.6 707.2 761.6 707.2 571.2 544 544 441.3 461.2 353.6 557.3 473.4 699.9 556.4 477.4 454.9 312.5 377.9 623.4 489.6 272] All rights reserved. Which of the following sequences follows this formula. 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. We will also give the Divergence Test for series in this section. If it converges, compute the limit. 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 Premium members get access to this practice exam along with our entire library of lessons taught by subject matter experts. We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. Contact us by phone at (877)266-4919, or by mail at 100ViewStreet#202, MountainView, CA94041. Images. Ratio test. (answer), Ex 11.1.5 Determine whether \(\left\{{n+47\over\sqrt{n^2+3n}}\right\}_{n=1}^{\infty}\) converges or diverges. x[[o6~cX/e`ElRm'1%J$%v)tb]1U2sRV}.l%s\Y UD+q}O+J %%EOF The practice tests are composed >> %PDF-1.5 % Solution. 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 >> 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 Our mission is to provide a free, world-class education to anyone, anywhere. /FontDescriptor 11 0 R (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 \(\sum_{n=0}^N {1\over e^n} + 10^{-4}\). Accessibility StatementFor more information contact us atinfo@libretexts.org. (answer). /FirstChar 0 It turns out the answer is no. bmkraft7. Good luck! Our mission is to provide a free, world-class education to anyone, anywhere. We also discuss differentiation and integration of power series. AP is a registered trademark of the College Board, which has not reviewed this resource. 816 816 272 299.2 489.6 489.6 489.6 489.6 489.6 792.7 435.2 489.6 707.2 761.6 489.6 Alternating series test. 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 (answer). 531.3 531.3 531.3] L7s[AQmT*Z;HK%H0yqt1r8 << endstream We will determine if a sequence in an increasing sequence or a decreasing sequence and hence if it is a monotonic sequence. (b) (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 \(\sum_{n=2}^N {1\over n(\ln n)^2} + 0.005\). /FontDescriptor 14 0 R Which rule represents the nth term in the sequence 9, 16, 23, 30? MULTIPLE CHOICE: Circle the best answer. /FirstChar 0 >> 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 >> Strip out the first 3 terms from the series \( \displaystyle \sum\limits_{n = 1}^\infty {\frac{{{2^{ - n}}}}{{{n^2} + 1}}} \). 772.4 811.3 431.9 541.2 833 666.2 947.3 784.1 748.3 631.1 775.5 745.3 602.2 573.9 833.3 833.3 833.3 833.3 1444.4 1277.8 555.6 1111.1 1111.1 1111.1 1111.1 1111.1 944.4 Here are a set of practice problems for the Series and Sequences chapter of the Calculus II notes. We also derive some well known formulas for Taylor series of \({\bf e}^{x}\) , \(\cos(x)\) and \(\sin(x)\) around \(x=0\). If you . /FontDescriptor 20 0 R Psychological Research & Experimental Design, All Teacher Certification Test Prep Courses. 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. Divergence Test. 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. 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 Level up on all the skills in this unit and collect up to 2000 Mastery points! (answer), Ex 11.2.3 Explain why \(\sum_{n=1}^\infty {3\over n}\) diverges. It turns out the answer is no. in calculus coursesincluding Calculus, Calculus II, Calculus III, AP Calculus and Precalculus. 21 0 obj 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 Which of the following is the 14th term of the sequence below? My calculus 2 exam on sequence, infinite series & power seriesThe exam: https://bit.ly/36OHYcsAll the convergence tests: https://bit.ly/2IzqokhBest friend an. (You may want to use Sage or a similar aid.) x=S0 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$&DN>+gt hb```9B 7N0$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@ " Alternating Series Test In this section we will discuss using the Alternating Series Test to determine if an infinite series converges or diverges. Determine whether the series converge or diverge. 2.(a). Each term is the sum of the previous two terms. In order to use either test the terms of the infinite series must be positive. (answer), Ex 11.9.3 Find a power series representation for \( 2/(1-x)^3\). (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 \( 10^{-3}\)? 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 . /Type/Font xu? ~k"xPeEV4Vcwww \ a:5d*%30EU9>,e92UU3Voj/$f BS!.eSloaY&h&Urm!U3L%g@'>`|$ogJ Consider the series n a n. Divergence Test: If lim n a n 0, then n a n diverges. You may also use any of these materials for practice. Series The Basics In this section we will formally define an infinite series. Other sets by this creator. Free Practice Test Instructions: Choose your answer to the question and click 'Continue' to see how you did. Power Series In this section we will give the definition of the power series as well as the definition of the radius of convergence and interval of convergence for a power series. 207 0 obj <> endobj >> Calculus II-Sequences and Series. << 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. For problems 1 3 perform an index shift so that the series starts at \(n = 3\). endstream endobj startxref To log in and use all the features of Khan Academy, please enable JavaScript in your browser. Published by Wiley. 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. /BaseFont/VMQJJE+CMR8 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. /Name/F5 /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 Ex 11.10.8 Find the first four terms of the Maclaurin series for \(\tan x\) (up to and including the \( x^3\) term). OR sequences are lists of numbers, where the numbers may or may not be determined by a pattern. Complementary General calculus exercises can be found for other Textmaps and can be accessed here. endobj Strip out the first 3 terms from the series n=1 2n n2 +1 n = 1 2 n n 2 + 1. Don't all infinite series grow to infinity? 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 Ex 11.1.2 Use the squeeze theorem to show that \(\lim_{n\to\infty} {n!\over n^n}=0\). 0 Let the factor without dx equal u and the factor with dx equal dv. Chapters include Linear Your instructor might use some of these in class.

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