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Calculus II - Sequences and Series Flashcards | Quizlet /Filter /FlateDecode (answer), Ex 11.2.6 Compute \(\sum_{n=0}^\infty {4^{n+1}\over 5^n}\). Determine whether the series converge or diverge. Each term is the product of the two previous terms. (answer), Ex 11.2.8 Compute \(\sum_{n=1}^\infty \left({3\over 5}\right)^n\). 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. Some infinite series converge to a finite value. 1000 1000 777.8 777.8 1000 1000 500 500 1000 1000 1000 777.8 1000 1000 611.1 611.1 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 xWKoFWlojCpP NDED$(lq"g|3g6X_&F1BXIM5d gOwaN9c,r|9
PDF FINAL EXAM CALCULUS 2 - Department of Mathematics %%EOF
,vEmO8/OuNVRaLPqB.*l. 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. 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. After each bounce, the ball reaches a height that is 2/3 of the height from which it previously fell. Donate or volunteer today! Course summary; . Then click 'Next Question' to answer the next question. /Length 1722 (answer), Ex 11.1.4 Determine whether \(\left\{{n^2+1\over (n+1)^2}\right\}_{n=0}^{\infty}\) converges or diverges. (1 point) Is the integral Z 1 1 1 x2 dx an improper integral? Applications of Series In this section we will take a quick look at a couple of applications of series. 272 761.6 462.4 462.4 761.6 734 693.4 707.2 747.8 666.2 639 768.3 734 353.2 503 761.2 531.3 590.3 560.8 414.1 419.1 413.2 590.3 560.8 767.4 560.8 560.8 472.2 531.3 1062.5 Ratio Test In this section we will discuss using the Ratio Test to determine if an infinite series converges absolutely or diverges. /LastChar 127 ZrNRG{I~(iw%0W5b)8*^
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What is the radius of convergence? At this time, I do not offer pdfs for solutions to individual problems. Complementary General calculus exercises can be found for other Textmaps and can be accessed here. 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. If it converges, compute the limit. 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 A brick wall has 60 bricks in the first row, but each row has 3 fewer bricks than the previous one. /LastChar 127 All other trademarks and copyrights are the property of their respective owners. Sequences can be thought of as functions whose domain is the set of integers. SAT Practice Questions- All Maths; SAT Practice Test Questions- Reading , Writing and Language; KS 1-2 Math, Science and SAT . 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. Then click 'Next Question' to answer the next question. 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 %PDF-1.5
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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. /Filter /FlateDecode All other trademarks and copyrights are the property of their respective owners. 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 Alternating series test. (answer), Ex 11.10.10 Use a combination of Maclaurin series and algebraic manipulation to find a series centered at zero for \( xe^{-x}\). Remark. 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 Derivatives, Integrals, Sequences & Series, and Vector Valued Functions. /FontDescriptor 8 0 R /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 may also use any of these materials for practice. 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 Then we can say that the series diverges without having to do any extra work. 31 terms. 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 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). Choose your answer to the question and click 'Continue' to see how you did. Remark. Here are a set of practice problems for the Series and Sequences chapter of the Calculus II notes. At this time, I do not offer pdf's for solutions to individual problems. Sequences and Series. YesNo 2.(b). 12 0 obj (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 \(\sum_{n=0}^N {1\over e^n} + 10^{-4}\). S.QBt'(d|/"XH4!qbnEriHX)Gs2qN/G jq8$$< For each function, find the Maclaurin series or Taylor series centered at $a$, and the radius of convergence. /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 /BaseFont/VMQJJE+CMR8 Ex 11.1.3 Determine whether {n + 47 n} . Learning Objectives. However, use of this formula does quickly illustrate how functions can be represented as a power series. Calculus II-Sequences and Series. Which equation below represents a geometric sequence? Chapters include Linear Choose your answer to the question and click 'Continue' to see how you did. (answer). !A1axw)}p]WgxmkFftu Good luck! Ex 11.7.3 Compute \(\lim_{n\to\infty} |a_n|^{1/n}\) for the series \(\sum 1/n^2\). x=S0 Harmonic series and p-series. 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). If it converges, compute the limit. Determine whether each series converges absolutely, converges conditionally, or diverges. << /Subtype/Type1
Infinite sequences and series | AP/College Calculus BC - Khan Academy Worksheets. 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. What is the sum of all the even integers from 2 to 250? 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. /Filter /FlateDecode
6.5E: Exercises for Comparison Test - Mathematics LibreTexts Determine whether the following series converge 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.
Calculus 2 | Math | Khan Academy Calculus (single and multi-variable) Ordinary Differential equations (upto 2nd order linear with Laplace transforms, including Dirac Delta functions and Fourier Series. We will also determine a sequence is bounded below, bounded above and/or bounded. /Filter /FlateDecode If it converges, compute the limit. Good luck!
Research Methods Midterm. The numbers used come from a sequence. 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. 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 . When you have completed the free practice test, click 'View Results' to see your results. (answer). endstream
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In order to use either test the terms of the infinite series must be positive. /BaseFont/PSJLQR+CMEX10 The sum of the steps forms an innite series, the topic of Section 10.2 and the rest of Chapter 10. Contact us by phone at (877)266-4919, or by mail at 100ViewStreet#202, MountainView, CA94041. Khan Academy is a 501(c)(3) nonprofit organization. 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. /LastChar 127 Which of the following is the 14th term of the sequence below? 750 750 750 1044.4 1044.4 791.7 791.7 583.3 583.3 638.9 638.9 638.9 638.9 805.6 805.6 507.9 433.7 395.4 427.7 483.1 456.3 346.1 563.7 571.2 589.1 483.8 427.7 555.4 505 Accessibility StatementFor more information contact us atinfo@libretexts.org. Series are sums of multiple terms. /FontDescriptor 23 0 R For problems 1 - 3 perform an index shift so that the series starts at n = 3 n = 3.
PDF M 172 - Calculus II - Chapter 10 Sequences and Series /Length 1247 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 Images. (b) Example 1. (answer). endobj 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 Khan Academy is a 501(c)(3) nonprofit organization. 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 A proof of the Integral Test is also given. 2 6 points 2.
Math 1242: Calculus II - University of North Carolina at Charlotte The chapter headings refer to Calculus, Sixth Edition by Hughes-Hallett et al. << 1 2 + 1 4 + 1 8 + = n=1 1 2n = 1 We will need to be careful, but it turns out that we can . Determine whether each series converges or diverges. 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 . /Type/Font 777.8 777.8] Ex 11.7.4 Compute \(\lim_{n\to\infty} |a_n|^{1/n}\) for the series \(\sum 1/n\). Each review chapter is packed with equations, formulas, and examples with solutions, so you can study smarter and score a 5! Free Practice Test Instructions: Choose your answer to the question and click 'Continue' to see how you did. Khan Academy is a nonprofit with the mission of providing a free, world-class education for anyone, anywhere. Ex 11.7.1 Compute \(\lim_{n\to\infty} |a_{n+1}/a_n|\) for the series \(\sum 1/n^2\). Each term is the sum of the previous two terms.
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. With an outline format that facilitates quick and easy review, Schaum's Outline of Calculus, Seventh Edition helps you understand basic concepts and get the extra practice you need to excel in these courses. /Subtype/Type1 About this unit. 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. /Name/F5 /FontDescriptor 17 0 R (answer), Ex 11.2.2 Explain why \(\sum_{n=1}^\infty {5\over 2^{1/n}+14}\) diverges. /BaseFont/UNJAYZ+CMR12 Learn how this is possible, how we can tell whether a series converges, and how we can explore convergence in Taylor and Maclaurin series. Sequences & Series in Calculus Chapter Exam. /FontDescriptor 11 0 R >> Note that some sections will have more problems than others and some will have more or less of a variety of problems. %PDF-1.5 (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. Series The Basics In this section we will formally define an infinite series. << Parametric equations, polar coordinates, and vector-valued functions Calculator-active practice: Parametric equations, polar coordinates, . In exercises 3 and 4, do not attempt to determine whether the endpoints are in the interval of convergence. 500 388.9 388.9 277.8 500 500 611.1 500 277.8 833.3 750 833.3 416.7 666.7 666.7 777.8 Ex 11.1.2 Use the squeeze theorem to show that limn n! << 531.3 531.3 531.3] Ex 11.9.5 Find a power series representation for \(\int\ln(1-x)\,dx\).
PDF Practice Problems Series & Sequences - MR. SOLIS' WEEBLY Choose the equation below that represents the rule for the nth term of the following geometric sequence: 128, 64, 32, 16, 8, . Choose your answer to the question and click 'Continue' to see how you did. 722.6 693.1 833.5 795.8 382.6 545.5 825.4 663.6 972.9 795.8 826.4 722.6 826.4 781.6 413.2 531.3 826.4 295.1 354.2 295.1 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 (answer), Ex 11.2.4 Compute \(\sum_{n=0}^\infty {4\over (-3)^n}- {3\over 3^n}\). 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 Each term is the difference of the previous two terms. Contact us by phone at (877)266-4919, or by mail at 100ViewStreet#202, MountainView, CA94041. Given that n=0 1 n3 +1 = 1.6865 n = 0 1 n 3 + 1 = 1.6865 determine the value of n=2 1 n3 +1 . /LastChar 127 Ex 11.10.8 Find the first four terms of the Maclaurin series for \(\tan x\) (up to and including the \( x^3\) term). (answer), Ex 11.2.5 Compute \(\sum_{n=0}^\infty {3\over 2^n}+ {4\over 5^n}\). 611.8 897.2 734 761.6 666.2 761.6 720.6 544 707.2 734 734 1006 734 734 598.4 272 Bottom line -- series are just a lot of numbers added together. To use integration by parts in Calculus, follow these steps: Decompose the entire integral (including dx) into two factors.
AP Calculus AB and BC: Chapter 9 -Infinite Sequences and Series : 9.4 . 777.8 444.4 444.4 444.4 611.1 777.8 777.8 777.8 777.8] copyright 2003-2023 Study.com. (You may want to use Sage or a similar aid.) /Name/F6
Math 106 (Calculus II): old exams | Mathematics | Bates College Don't all infinite series grow to infinity? MULTIPLE CHOICE: Circle the best answer. (answer), Ex 11.9.3 Find a power series representation for \( 2/(1-x)^3\). /FirstChar 0 hbbd```b``~"A$"
"Y`L6`RL,-`sA$w64= f[" RLMu;@jAl[`3H^Ne`?$4 These are homework exercises to accompany David Guichard's "General Calculus" Textmap. When you have completed the free practice test, click 'View Results' to see your results. Find the radius and interval of convergence for each series. (answer), Ex 11.2.7 Compute \(\sum_{n=0}^\infty {3^{n+1}\over 7^{n+1}}\). Series Infinite geometric series: Series nth-term test: Series Integral test: Series Harmonic series and p-series: Series Comparison tests: .
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Solving My Calc 2 Exam#3 (Sequence, Infinite Series & Power Series) 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. 666.7 1000 1000 1000 1000 1055.6 1055.6 1055.6 777.8 666.7 666.7 450 450 450 450 Which is the infinite sequence starting with 1 where each number is the previous number times 3? 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 We will also see how we can use the first few terms of a power series to approximate a function. 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 For problems 1 3 perform an index shift so that the series starts at \(n = 3\). Comparison Test: This applies . 18 0 obj % 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 . (answer). At this time, I do not offer pdf's for . /Subtype/Type1 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 /Length 569
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PDF Ap Calculus Ab Bc Kelley Copy - gny.salvationarmy.org