(answer). 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 << All other trademarks and copyrights are the property of their respective owners. Which of the following represents the distance the ball bounces from the first to the seventh bounce with sigma notation? /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 Remark. We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. Indiana Core Assessments Mathematics: Test Prep & Study Guide. SAT Practice Questions- All Maths; SAT Practice Test Questions- Reading , Writing and Language; KS 1-2 Math, Science and SAT . Sequences and Series. /Name/F2 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. 666.7 1000 1000 1000 1000 1055.6 1055.6 1055.6 777.8 666.7 666.7 450 450 450 450 Which of the following is the 14th term of the sequence below? 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. If it converges, compute the limit. Then we can say that the series diverges without having to do any extra work. PDF FINAL EXAM CALCULUS 2 - Department of Mathematics We will also see how we can use the first few terms of a power series to approximate a function. To use the Geometric Series formula, the function must be able to be put into a specific form, which is often impossible. /FirstChar 0 Choose the equation below that represents the rule for the nth term of the following geometric sequence: 128, 64, 32, 16, 8, . /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 x=S0 /LastChar 127 665 570.8 924.4 812.6 568.1 670.2 380.8 380.8 380.8 979.2 979.2 410.9 514 416.3 421.4 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. For each of the following series, determine which convergence test is the best to use and explain why. 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Each term is the difference of the previous two terms. Determine whether each series converges absolutely, converges conditionally, or diverges. 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. << Then click 'Next Question' to answer the next question. 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 endobj /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 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. Ex 11.7.4 Compute \(\lim_{n\to\infty} |a_n|^{1/n}\) for the series \(\sum 1/n\). /Type/Font Given item A, which of the following would be the value of item B? xu? ~k"xPeEV4Vcwww \ a:5d*%30EU9>,e92UU3Voj/$f
BS!.eSloaY&h&Urm!U3L%g@'>`|$ogJ (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 \(\sum_{n=0}^N {1\over e^n} + 10^{-4}\). Complementary General calculus exercises can be found for other Textmaps and can be accessed here. Premium members get access to this practice exam along with our entire library of lessons taught by subject matter experts. /Filter /FlateDecode /Type/Font 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 /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 (5 points) Evaluate the integral: Z 1 1 1 x2 dx = SOLUTION: The function 1/x2 is undened at x = 0, so we we must evaluate the im- proper integral as a limit. All rights reserved. stream Absolute and conditional convergence. Applications of Series In this section we will take a quick look at a couple of applications of series. endobj Integral Test: If a n = f ( n), where f ( x) is a non-negative non-increasing function, then. If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. 17 0 obj Here are a set of practice problems for the Series and Sequences chapter of the Calculus II notes. 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. 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 Series | Calculus 2 | Math | Khan Academy In other words, a series is the sum of a sequence. Don't all infinite series grow to infinity? 777.8 444.4 444.4 444.4 611.1 777.8 777.8 777.8 777.8] 26 0 obj 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. 18 0 obj >> (answer), Ex 11.10.9 Use a combination of Maclaurin series and algebraic manipulation to find a series centered at zero for \( x\cos (x^2)\). 31 terms. (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 \(\sum_{n=1}^N {\ln n\over n^2} + 0.005\). Note that some sections will have more problems than others and some will have more or less of a variety of problems. Then click 'Next Question' to answer the . Images. Ex 11.8.1 \(\sum_{n=0}^\infty n x^n\) (answer), Ex 11.8.2 \(\sum_{n=0}^\infty {x^n\over n! Learn how this is possible, how we can tell whether a series converges, and how we can explore convergence in Taylor and Maclaurin series. 816 816 272 299.2 489.6 489.6 489.6 489.6 489.6 792.7 435.2 489.6 707.2 761.6 489.6 (answer). }\) (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! Contact us by phone at (877)266-4919, or by mail at 100ViewStreet#202, MountainView, CA94041. Here are a set of practice problems for the Series and Sequences chapter of the Calculus II notes. /FirstChar 0 endstream
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Ex 11.10.8 Find the first four terms of the Maclaurin series for \(\tan x\) (up to and including the \( x^3\) term). Khan Academy is a nonprofit with the mission of providing a free, world-class education for anyone, anywhere. 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? 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. Which of the following sequences follows this formula. (answer), Ex 11.1.4 Determine whether \(\left\{{n^2+1\over (n+1)^2}\right\}_{n=0}^{\infty}\) converges or diverges. /Length 2492 Ex 11.1.2 Use the squeeze theorem to show that \(\lim_{n\to\infty} {n!\over n^n}=0\). Quiz 2: 8 questions Practice what you've learned, and level up on the above skills. >> What is the sum of all the even integers from 2 to 250? << Complementary General calculus exercises can be found for other Textmaps and can be accessed here. Our mission is to provide a free, world-class education to anyone, anywhere. Chapters include Linear Infinite sequences and series | AP/College Calculus BC - Khan Academy 8 0 obj Which rule represents the nth term in the sequence 9, 16, 23, 30? 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. << 1 2 + 1 4 + 1 8 + = n=1 1 2n = 1 We will need to be careful, but it turns out that we can . << %PDF-1.5 ,vEmO8/OuNVRaLPqB.*l. MATH 126 Medians and Such. Your instructor might use some of these in class. Infinite series are sums of an infinite number of 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. web manual for algebra 2 and pre calculus volume ii pre calculus for dummies jan 20 2021 oers an introduction to the principles of pre calculus covering such topics as functions law of sines and cosines identities sequences series and binomials algebra 2 homework practice workbook oct 29 2021 algebra ii practice tests varsity tutors - Nov 18 . (answer), Ex 11.9.3 Find a power series representation for \( 2/(1-x)^3\). x[[o6~cX/e`ElRm'1%J$%v)tb]1U2sRV}.l%s\Y UD+q}O+J Good luck! For problems 1 3 perform an index shift so that the series starts at \(n = 3\). 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. Then click 'Next Question' to answer the next question. 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 Ex 11.1.3 Determine whether \(\{\sqrt{n+47}-\sqrt{n}\}_{n=0}^{\infty}\) converges or diverges. 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. n = 1 n 2 + 2 n n 3 + 3 n . 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 Most sections should have a range of difficulty levels in the problems although this will vary from section to section. 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 endobj We will examine Geometric Series, Telescoping Series, and Harmonic Series. Learn for free about math, art, computer programming, economics, physics, chemistry, biology, medicine, finance, history, and more. Legal. Ratio Test In this section we will discuss using the Ratio Test to determine if an infinite series converges absolutely or diverges. 979.2 979.2 979.2 272 272 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 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. >> Integral test. 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 Free Practice Test Instructions: Choose your answer to the question and click 'Continue' to see how you did. 21 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 A proof of the Root Test is also given. (answer). Some infinite series converge to a finite value. &/ r
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. 531.3 531.3 531.3] 12 0 obj /LastChar 127 %%EOF
What if the interval is instead \([1,3/2]\)? % in calculus coursesincluding Calculus, Calculus II, Calculus III, AP Calculus and Precalculus. Determine whether each series converges or diverges. Level up on all the skills in this unit and collect up to 2000 Mastery points! 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 sCA%HGEH[ Ah)lzv<7'9&9X}xbgY[ xI9i,c_%tz5RUam\\6(ke9}Yv`B7yYdWrJ{KZVUYMwlbN_>[wle\seUy24P,PyX[+l\c $w^rvo]cYc@bAlfi6);;wOF&G_. %|S#?\A@D-oS)lW=??nn}y]Tb!!o_=;]ha,J[. Series are sums of multiple terms. 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 Here are a set of practice problems for the Series and Sequences chapter of the Calculus II notes. 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 1062.5 833.3 833.3 833.3 833.3 1444.4 1277.8 555.6 1111.1 1111.1 1111.1 1111.1 1111.1 944.4 Bottom line -- series are just a lot of numbers added together. n a n converges if and only if the integral 1 f ( x) d x converges. /LastChar 127 We will determine if a sequence in an increasing sequence or a decreasing sequence and hence if it is a monotonic sequence. Sequences can be thought of as functions whose domain is the set of integers. The numbers used come from a sequence. stream Section 10.3 : Series - Basics. /FontDescriptor 23 0 R The steps are terms in the sequence. (answer), Ex 11.1.6 Determine whether \(\left\{{2^n\over n! Question 5 5. The Integral Test can be used on a infinite series provided the terms of the series are positive and decreasing. Alternating Series Test - In this section we will discuss using the Alternating Series Test to determine if an infinite series converges or diverges. Calculus II - Series & Sequences (Practice Problems) - Lamar University )^2\over n^n}(x-2)^n\) (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 \(\ln 2\). Which of the following sequences is NOT a geometric sequence? /Subtype/Type1 /FirstChar 0 copyright 2003-2023 Study.com. 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! Determine whether the series is convergent or divergent. endstream /FirstChar 0 Research Methods Midterm. If you're seeing this message, it means we're having trouble loading external resources on our website. << Let the factor without dx equal u and the factor with dx equal dv. /BaseFont/BPHBTR+CMMI12 n = 1 n2 + 2n n3 + 3n2 + 1. (b) )^2\over n^n}\) (answer). 68 0 obj Choose your answer to the question and click 'Continue' to see how you did. Learn how this is possible, how we can tell whether a series converges, and how we can explore convergence in . Determine whether the sequence converges or diverges. PDF Review Sheet for Calculus 2 Sequences and Series - Derrick Chung Part II. Ratio test. (answer), Ex 11.2.7 Compute \(\sum_{n=0}^\infty {3^{n+1}\over 7^{n+1}}\). /Name/F6 /Length 1247 A proof of the Ratio Test is also given. If it con-verges, nd the limit. S.QBt'(d|/"XH4!qbnEriHX)Gs2qN/G jq8$$< To use integration by parts in Calculus, follow these steps: Decompose the entire integral (including dx) into two factors. Which of the following is the 14th term of the sequence below? Calculus II For Dummies Cheat Sheet - dummies The Alternating Series Test can be used only if the terms of the series alternate in sign. Donate or volunteer today! Sequences and Numerical series. Series The Basics In this section we will formally define an infinite series. We will also give many of the basic facts, properties and ways we can use to manipulate a series. endstream
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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 also derive some well known formulas for Taylor series of \({\bf e}^{x}\) , \(\cos(x)\) and \(\sin(x)\) around \(x=0\). (answer), Ex 11.2.3 Explain why \(\sum_{n=1}^\infty {3\over n}\) diverges. It turns out the answer is no. A proof of the Integral Test is also given. << Comparison Test: This applies . 2.(a). (answer), Ex 11.4.6 Approximate \(\sum_{n=1}^\infty (-1)^{n-1}{1\over n^4}\) to two decimal places. endstream /FontDescriptor 14 0 R 979.2 489.6 489.6 489.6] AP Calculus AB and BC: Chapter 9 -Infinite Sequences and Series : 9.4 Derivatives, Integrals, Sequences & Series, and Vector Valued Functions. 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 . (answer), Ex 11.2.5 Compute \(\sum_{n=0}^\infty {3\over 2^n}+ {4\over 5^n}\). Calculus 2 | Math | Khan Academy Calculus II - Series & Sequences (Practice Problems) - Lamar University 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 Math 106 (Calculus II): old exams | Mathematics | Bates College (answer), Ex 11.2.9 Compute \(\sum_{n=1}^\infty {3^n\over 5^{n+1}}\). 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 /Type/Font If it converges, compute the limit. /Name/F4 (answer). stream /Subtype/Type1 More on Sequences In this section we will continue examining sequences. 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. 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 Then determine if the series converges or diverges. 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. 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 These are homework exercises to accompany David Guichard's "General Calculus" Textmap. For each function, find the Maclaurin series or Taylor series centered at $a$, and the radius of convergence. 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. >> /FontDescriptor 20 0 R
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