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Determine whether the following series converge or diverge. Then determine if the series converges or diverges. Strip out the first 3 terms from the series n=1 2n n2 +1 n = 1 2 n n 2 + 1. 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). /FirstChar 0 Level up on all the skills in this unit and collect up to 2000 Mastery points! If a geometric series begins with the following term, what would the next term be? 9.8 Power Series Chapter 9 Sequences and Series Calculus II 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] PDF Arithmetic Sequences And Series Practice Problems 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 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. Each term is the product of the two previous terms. A brick wall has 60 bricks in the first row, but each row has 3 fewer bricks than the previous one. << When you have completed the free practice test, click 'View Results' to see your results. Math Journey: Calculus, ODEs, Linear Algebra and Beyond 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. What is the radius of convergence? It turns out the answer is no. Given item A, which of the following would be the value of item B? Infinite series are sums of an infinite number of terms. Calculus II - Series - The Basics (Practice Problems) - Lamar University If you . For problems 1 3 perform an index shift so that the series starts at \(n = 3\). What if the interval is instead \([1,3/2]\)? To use integration by parts in Calculus, follow these steps: Decompose the entire integral (including dx) into two factors. /Name/F4 /Length 465 >> /FirstChar 0 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. Which of the following sequences is NOT a geometric sequence? Strip out the first 3 terms from the series \( \displaystyle \sum\limits_{n = 1}^\infty {\frac{{{2^{ - n}}}}{{{n^2} + 1}}} \). Research Methods Midterm. Some infinite series converge to a finite value. 62 0 obj /LastChar 127 /BaseFont/CQGOFL+CMSY10 UcTIjeB#vog-TM'FaTzG(:k-BNQmbj}'?^h<=XgS/]o4Ilv%Jm (answer), Ex 11.2.5 Compute \(\sum_{n=0}^\infty {3\over 2^n}+ {4\over 5^n}\). << Some infinite series converge to a finite value. }\) (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! /Type/Font /Filter /FlateDecode Integral test. xu? ~k"xPeEV4Vcwww \ a:5d*%30EU9>,e92UU3Voj/$f
BS!.eSloaY&h&Urm!U3L%g@'>`|$ogJ 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. (answer), Ex 11.2.3 Explain why \(\sum_{n=1}^\infty {3\over n}\) diverges. 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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), \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}}}\) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( 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If it converges, compute the limit. (answer). Donate or volunteer today! endobj Worked example: sequence convergence/divergence, Partial sums: formula for nth term from partial sum, Partial sums: term value from partial sum, Worked example: convergent geometric series, Worked example: divergent geometric series, Infinite geometric series word problem: bouncing ball, Infinite geometric series word problem: repeating decimal, Proof of infinite geometric series formula, Convergent & divergent geometric series (with manipulation), Level up on the above skills and collect up to 320 Mastery points, Determine absolute or conditional convergence, Level up on the above skills and collect up to 640 Mastery points, Worked example: alternating series remainder, Taylor & Maclaurin polynomials intro (part 1), Taylor & Maclaurin polynomials intro (part 2), Worked example: coefficient in Maclaurin polynomial, Worked example: coefficient in Taylor polynomial, Visualizing Taylor polynomial approximations, Worked example: estimating sin(0.4) using Lagrange error bound, Worked example: estimating e using Lagrange error bound, Worked example: cosine function from power series, Worked example: recognizing function from Taylor series, Maclaurin series of sin(x), cos(x), and e, Finding function from power series by integrating, Integrals & derivatives of functions with known power series, Interval of convergence for derivative and integral, Converting explicit series terms to summation notation, Converting explicit series terms to summation notation (n 2), Formal definition for limit of a sequence, Proving a sequence converges using the formal definition, Infinite geometric series formula intuition, Proof of infinite geometric series as a limit. PDF Calc II: Practice Final Exam - Columbia University \ _* %l~G"tytO(J*l+X@ uE: m/ ~&Q24Nss(7F!ky=4 Mijo8t;v Then click 'Next Question' to answer the next question. (answer), Ex 11.2.4 Compute \(\sum_{n=0}^\infty {4\over (-3)^n}- {3\over 3^n}\). 21 0 obj The following is a list of worksheets and other materials related to Math 129 at the UA. >> (answer), Ex 11.2.9 Compute \(\sum_{n=1}^\infty {3^n\over 5^{n+1}}\). 5.3.3 Estimate the value of a series by finding bounds on its remainder term. In exercises 3 and 4, do not attempt to determine whether the endpoints are in the interval of convergence. 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. Let the factor without dx equal u and the factor with dx equal dv. 70 terms. /LastChar 127 We will focus on the basic terminology, limits of sequences and convergence of sequences in this section. Quiz 2: 8 questions Practice what you've learned, and level up on the above skills. xTn0+,ITi](N@ fH2}W"UG'.% Z#>y{!9kJ+ %%EOF
/Length 569 888.9 888.9 888.9 888.9 888.9 888.9 888.9 666.7 875 875 875 875 611.1 611.1 833.3 (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 \(\sum_{n=2}^N {1\over n(\ln n)^2} + 0.005\). /Subtype/Type1 /FontDescriptor 17 0 R /BaseFont/SFGTRF+CMSL12 9 0 obj 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. stream 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. (answer), Ex 11.10.10 Use a combination of Maclaurin series and algebraic manipulation to find a series centered at zero for \( xe^{-x}\). (answer), Ex 11.9.2 Find a power series representation for \(1/(1-x)^2\). copyright 2003-2023 Study.com. Each review chapter is packed with equations, formulas, and examples with solutions, so you can study smarter and score a 5! (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 \( 10^{-3}\)? /Length 2492 Series Infinite geometric series: Series nth-term test: Series Integral test: Series Harmonic series and p-series: Series Comparison tests: . 11.E: Sequences and Series (Exercises) - Mathematics LibreTexts Here are a set of practice problems for the Series and Sequences chapter of the Calculus II notes. We will also give many of the basic facts, properties and ways we can use to manipulate a series. The sum of the steps forms an innite series, the topic of Section 10.2 and the rest of Chapter 10. 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. Psychological Research & Experimental Design, All Teacher Certification Test Prep Courses. 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 Then click 'Next Question' to answer the next question. 5.3.2 Use the integral test to determine the convergence of a series. /FirstChar 0 At this time, I do not offer pdfs for solutions to individual problems. << nth-term test. /Name/F3 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 (answer), Ex 11.9.3 Find a power series representation for \( 2/(1-x)^3\). )^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 \(\ln 2\). Ex 11.1.3 Determine whether \(\{\sqrt{n+47}-\sqrt{n}\}_{n=0}^{\infty}\) converges or diverges. >> 2.(a). Which of the following is the 14th term of the sequence below? 15 0 obj Indiana Core Assessments Mathematics: Test Prep & Study Guide. Series The Basics In this section we will formally define an infinite series. << 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 raVQ1CKD3`
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`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 (answer), Ex 11.2.6 Compute \(\sum_{n=0}^\infty {4^{n+1}\over 5^n}\). 777.8 777.8] /BaseFont/VMQJJE+CMR8 Good luck! PDF Review Sheet for Calculus 2 Sequences and Series - Derrick Chung Khan Academy is a nonprofit with the mission of providing a free, world-class education for anyone, anywhere. /Length 1722 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. copyright 2003-2023 Study.com. Determine whether the sequence converges or diverges. 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? The book contains eight practice tests five practice tests for Calculus AB and three practice tests for Calculus BC. Martha_Austin Teacher. Which of the following represents the distance the ball bounces from the first to the seventh bounce with sigma notation? 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. Which of the following is the 14th term of the sequence below? << xWKoFWlojCpP NDED$(lq"g|3g6X_&F1BXIM5d gOwaN9c,r|9 (You may want to use Sage or a similar aid.) Ex 11.7.2 Compute \(\lim_{n\to\infty} |a_{n+1}/a_n|\) for the series \(\sum 1/n\). Ex 11.7.9 Prove theorem 11.7.3, the root test. /Name/F5 Images. 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. Our mission is to provide a free, world-class education to anyone, anywhere. /Widths[606.7 816 748.3 679.6 728.7 811.3 765.8 571.2 652.8 598 757.6 622.8 552.8 979.2 489.6 489.6 489.6] A proof of the Ratio Test is also given. Consider the series n a n. Divergence Test: If lim n a n 0, then n a n diverges. /Subtype/Type1 % (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 \(\sum_{n=0}^N {1\over e^n} + 10^{-4}\). All other trademarks and copyrights are the property of their respective owners. We will examine Geometric Series, Telescoping Series, and Harmonic Series. 21 terms. Math 106 (Calculus II): old exams | Mathematics | Bates College All rights reserved. The chapter headings refer to Calculus, Sixth Edition by Hughes-Hallett et al. 4 avwo/MpLv)
_C>5p*)i=^m7eE. 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. PDF Practice Problems Series & Sequences - MR. SOLIS' WEEBLY %PDF-1.5 << At this time, I do not offer pdf's for . Choosing a Convergence Test | Calculus II - Lumen Learning }\right\}_{n=0}^{\infty}\) converges or diverges. Determine whether the series is convergent or divergent. (answer), Ex 11.2.2 Explain why \(\sum_{n=1}^\infty {5\over 2^{1/n}+14}\) diverges. 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 Note that some sections will have more problems than others and some will have more or less of a variety of problems. (answer), Ex 11.2.7 Compute \(\sum_{n=0}^\infty {3^{n+1}\over 7^{n+1}}\). /FontDescriptor 8 0 R YesNo 2.(b). Solution. Which of the following sequences is NOT a geometric sequence? We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. Quiz 1: 5 questions Practice what you've learned, and level up on the above skills. PDF Ap Calculus Ab Bc Kelley Copy - gny.salvationarmy.org /LastChar 127 (answer), Ex 11.2.8 Compute \(\sum_{n=1}^\infty \left({3\over 5}\right)^n\). 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 1000 1000 777.8 777.8 1000 1000 500 500 1000 1000 1000 777.8 1000 1000 611.1 611.1 endobj 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 Ex 11.10.8 Find the first four terms of the Maclaurin series for \(\tan x\) (up to and including the \( x^3\) term). 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. 666.7 1000 1000 1000 1000 1055.6 1055.6 1055.6 777.8 666.7 666.7 450 450 450 450 When you have completed the free practice test, click 'View Results' to see your results. 722.2 777.8 777.8 611.1 798.5 656.8 526.5 771.4 527.8 718.7 594.9 844.5 544.5 677.8 826.4 531.3 958.7 1076.8 826.4 295.1 295.1 531.3 885.4 531.3 885.4 826.4 295.1 413.2 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. S.QBt'(d|/"XH4!qbnEriHX)Gs2qN/G jq8$$< 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.
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