(answer), Ex 11.9.3 Find a power series representation for \( 2/(1-x)^3\). endobj Root Test In this section we will discuss using the Root Test to determine if an infinite series converges absolutely or diverges. The chapter headings refer to Calculus, Sixth Edition by Hughes-Hallett et al. Integral test. Level up on all the skills in this unit and collect up to 2000 Mastery points! % (You may want to use Sage or a similar aid.) Sequences review (practice) | Series | Khan Academy 18 0 obj Parametric equations, polar coordinates, and vector-valued functions Calculator-active practice: Parametric equations, polar coordinates, . To log in and use all the features of Khan Academy, please enable JavaScript in your browser. n = 1 n2 + 2n n3 + 3n2 + 1. Ex 11.1.3 Determine whether \(\{\sqrt{n+47}-\sqrt{n}\}_{n=0}^{\infty}\) converges or diverges. Which is the infinite sequence starting with 1 where each number is the previous number times 3? PDF Calculus II Series - Things to Consider - California State University The Integral Test can be used on a infinite series provided the terms of the series are positive and decreasing. Applications of Series In this section we will take a quick look at a couple of applications of series. 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 Which of the following sequences is NOT a geometric sequence? Calculus II For Dummies Cheat Sheet - dummies PDF Schaums Outline Of Differential Equations 4th Edition Schaums Outline However, use of this formula does quickly illustrate how functions can be represented as a power series. Which of the following sequences follows this formula? /FirstChar 0 Don't all infinite series grow to infinity? 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. << Math Journey: Calculus, ODEs, Linear Algebra and Beyond Series Infinite geometric series: Series nth-term test: Series Integral test: Series Harmonic series and p-series: Series Comparison tests: . If it converges, compute the limit. << 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. 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 We will also give many of the basic facts, properties and ways we can use to manipulate a series. What if the interval is instead \([1,3/2]\)? 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. (answer), Ex 11.2.5 Compute \(\sum_{n=0}^\infty {3\over 2^n}+ {4\over 5^n}\). AP Calculus AB and BC: Chapter 9 -Infinite Sequences and Series : 9.4 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. 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 11.6.1 \(\sum_{n=1}^\infty (-1)^{n-1}{1\over 2n^2+3n+5}\) (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). Don't all infinite series grow to infinity? 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. Khan Academy is a nonprofit with the mission of providing a free, world-class education for anyone, anywhere. PDF Arithmetic Sequences And Series Practice Problems 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. Calculus II - Series & Sequences (Practice Problems) - Lamar University /BaseFont/BPHBTR+CMMI12 |: The Ratio Test shows us that regardless of the choice of x, the series converges. >> 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. Research Methods Midterm. Here are a set of practice problems for the Series and Sequences chapter of the Calculus II notes. 1 2 + 1 4 + 1 8 + = n=1 1 2n = 1 We will need to be careful, but it turns out that we can . << Which is the finite sequence of four multiples of 9, starting with 9? >> 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 After each bounce, the ball reaches a height that is 2/3 of the height from which it previously fell. Premium members get access to this practice exam along with our entire library of lessons taught by subject matter experts. We will examine Geometric Series, Telescoping Series, and Harmonic Series. 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] If the series is an alternating series, determine whether it converges absolutely, converges conditionally, or diverges. 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). Complementary General calculus exercises can be found for other Textmaps and can be accessed here. For each function, find the Maclaurin series or Taylor series centered at $a$, and the radius of convergence. /Name/F6 611.8 897.2 734 761.6 666.2 761.6 720.6 544 707.2 734 734 1006 734 734 598.4 272 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. We also discuss differentiation and integration of power series. (answer). If you . xTn0+,ITi](N@ fH2}W"UG'.% Z#>y{!9kJ+ 979.2 489.6 489.6 489.6] 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. Remark. Learning Objectives. 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 We will focus on the basic terminology, limits of sequences and convergence of sequences in this section. 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. Mediansandsuch - Medians - MATH 126 Medians and Such Let X be - Studocu PDF Practice Problems Series & Sequences - MR. SOLIS' WEEBLY /FirstChar 0 More on Sequences In this section we will continue examining sequences. 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 \(\sum_{n=1}^N {1\over n^4} + 0.005\). UcTIjeB#vog-TM'FaTzG(:k-BNQmbj}'?^h<=XgS/]o4Ilv%Jm Ex 11.7.1 Compute \(\lim_{n\to\infty} |a_{n+1}/a_n|\) for the series \(\sum 1/n^2\). Let the factor without dx equal u and the factor with dx equal dv. (answer), Ex 11.2.6 Compute \(\sum_{n=0}^\infty {4^{n+1}\over 5^n}\). The numbers used come from a sequence. (answer), Ex 11.2.2 Explain why \(\sum_{n=1}^\infty {5\over 2^{1/n}+14}\) diverges. 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. Complementary General calculus exercises can be found for other Textmaps and can be accessed here. 26 0 obj in calculus coursesincluding Calculus, Calculus II, Calculus III, AP Calculus and Precalculus. /Name/F2 (answer), Ex 11.3.11 Find an \(N\) so that \(\sum_{n=1}^\infty {\ln n\over n^2}\) is between \(\sum_{n=1}^N {\ln n\over n^2}\) and \(\sum_{n=1}^N {\ln n\over n^2} + 0.005\). A proof of the Ratio Test is also given. 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 /Type/Font Each term is the product of the two previous terms. 31 terms. Ratio Test In this section we will discuss using the Ratio Test to determine if an infinite series converges absolutely or diverges. Infinite sequences and series | AP/College Calculus BC - Khan Academy 15 0 obj Calculus II - Series - The Basics (Practice Problems) - Lamar University Sequences & Series in Calculus Chapter Exam. Calculus II-Sequences and Series. /LastChar 127 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. /Filter /FlateDecode (answer), Ex 11.10.9 Use a combination of Maclaurin series and algebraic manipulation to find a series centered at zero for \( x\cos (x^2)\). (answer), Ex 11.9.4 Find a power series representation for \( 1/(1-x)^3\). << endobj 9 0 obj \ _* %l~G"tytO(J*l+X@ uE: m/ ~&Q24Nss(7F!ky=4 Mijo8t;v 70 terms. /Widths[777.8 277.8 777.8 500 777.8 500 777.8 777.8 777.8 777.8 777.8 777.8 777.8 590.3 767.4 795.8 795.8 1091 795.8 795.8 649.3 295.1 531.3 295.1 531.3 295.1 295.1 Ex 11.7.4 Compute \(\lim_{n\to\infty} |a_n|^{1/n}\) for the series \(\sum 1/n\). Solving My Calc 2 Exam#3 (Sequence, Infinite Series & Power 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 A Lot of Series Test Practice Problems - YouTube Then click 'Next Question' to answer the next question. 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 We will also give many of the basic facts and properties well need as we work with sequences. /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 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. 6.5E: Exercises for Comparison Test - Mathematics LibreTexts 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 Calculus II - Sequences and Series Flashcards | Quizlet 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 Series The Basics In this section we will formally define an infinite series. When you have completed the free practice test, click 'View Results' to see your results. Course summary; . %PDF-1.5 % What is the 83rd term of the sequence 91, 87, 83, 79, ( = a. Ratio test. (answer), Ex 11.2.4 Compute \(\sum_{n=0}^\infty {4\over (-3)^n}- {3\over 3^n}\). endobj 45 0 obj (answer), Ex 11.1.4 Determine whether \(\left\{{n^2+1\over (n+1)^2}\right\}_{n=0}^{\infty}\) converges or diverges. 1000 1000 777.8 777.8 1000 1000 500 500 1000 1000 1000 777.8 1000 1000 611.1 611.1 (a) $\sum_{n=1}^{\infty} \frac{(-1)^n}{\sqrt{n}}$ (b) $\sum_{n=1}^{\infty}(-1)^n \frac{n}{2 n-1}$ 5.3 The Divergence and Integral Tests - Calculus Volume 2 - OpenStax All other trademarks and copyrights are the property of their respective owners. (answer), Ex 11.2.1 Explain why \(\sum_{n=1}^\infty {n^2\over 2n^2+1}\) diverges. 777.8 777.8] /Length 569 }\right\}_{n=0}^{\infty}\) converges or diverges. /FirstChar 0 /Subtype/Type1 Calculus (single and multi-variable) Ordinary Differential equations (upto 2nd order linear with Laplace transforms, including Dirac Delta functions and Fourier Series. Sequences and Series for Calculus Chapter Exam - Study.com Alternating series test - Wikipedia Donate or volunteer today! )Ltgx?^eaT'&+n+hN4*D^UR!8UY@>LqS%@Cp/-12##DR}miBw6"ja+WjU${IH$5j!j-I1 531.3 531.3 531.3 295.1 295.1 295.1 826.4 501.7 501.7 826.4 795.8 752.1 767.4 811.1 xYKs6W(MCG:9iIO=(lkFRI$x$AMN/" J?~i~d cXf9o/r.&Lxy%/D-Yt+"LX]Sfp]Xl-aM_[6(*~mQbh*28AjZx0 =||. Which of the following represents the distance the ball bounces from the first to the seventh bounce with sigma notation? It turns out the answer is no. /FontDescriptor 14 0 R Proofs for both tests are also given. 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 /FirstChar 0 My calculus 2 exam on sequence, infinite series & power seriesThe exam: https://bit.ly/36OHYcsAll the convergence tests: https://bit.ly/2IzqokhBest friend an. }\) (answer), Ex 11.8.3 \(\sum_{n=1}^\infty {n!\over n^n}x^n\) (answer), Ex 11.8.4 \(\sum_{n=1}^\infty {n!\over n^n}(x-2)^n\) (answer), Ex 11.8.5 \(\sum_{n=1}^\infty {(n! 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.

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