>> Your instructor might use some of these in class. Course summary; . Images. We will also see how we can use the first few terms of a power series to approximate a function. 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 A proof of the Alternating Series Test is also given. More on Sequences In this section we will continue examining sequences. When you have completed the free practice test, click 'View Results' to see your results. Psychological Research & Experimental Design, All Teacher Certification Test Prep Courses. /FontDescriptor 8 0 R >> Ex 11.7.4 Compute \(\lim_{n\to\infty} |a_n|^{1/n}\) for the series \(\sum 1/n\). (answer). 31 terms. 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. Applications of Series In this section we will take a quick look at a couple of applications of series. 70 terms. (answer), Ex 11.2.5 Compute \(\sum_{n=0}^\infty {3\over 2^n}+ {4\over 5^n}\). /Name/F6 \ _* %l~G"tytO(J*l+X@ uE: m/ ~&Q24Nss(7F!ky=4 Mijo8t;v /FirstChar 0 Accessibility StatementFor more information contact us atinfo@libretexts.org. xu? ~k"xPeEV4Vcwww \ a:5d*%30EU9>,e92UU3Voj/$f BS!.eSloaY&h&Urm!U3L%g@'>`|$ogJ Chapters include Linear endstream endobj startxref Quiz 1: 5 questions Practice what you've learned, and level up on the above skills. When you have completed the free practice test, click 'View Results' to see your results. 0 Series Infinite geometric series: Series nth-term test: Series Integral test: Series Harmonic series and p-series: Series Comparison tests: . Choosing a Convergence Test | Calculus II - Lumen Learning 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] 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. YesNo 2.(b). (answer), Ex 11.1.6 Determine whether \(\left\{{2^n\over n! Then click 'Next Question' to answer the next question. Ex 11.1.2 Use the squeeze theorem to show that \(\lim_{n\to\infty} {n!\over n^n}=0\). 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 . 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. >> 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 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. 18 0 obj It turns out the answer is no. nn = 0. )^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 \(\ln 2\). &/ r 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. Bottom line -- series are just a lot of numbers added together. Determine whether each series converges or diverges. 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. Which one of these sequences is a finite sequence? 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 in calculus coursesincluding Calculus, Calculus II, Calculus III, AP Calculus and Precalculus. 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 (answer), Ex 11.2.2 Explain why \(\sum_{n=1}^\infty {5\over 2^{1/n}+14}\) diverges. 611.8 897.2 734 761.6 666.2 761.6 720.6 544 707.2 734 734 1006 734 734 598.4 272 A proof of the Root Test is also given. UcTIjeB#vog-TM'FaTzG(:k-BNQmbj}'?^h<=XgS/]o4Ilv%Jm Series are sums of multiple terms. 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, \( \displaystyle \sum\limits_{n = 1}^\infty {\left( {n{2^n} - {3^{1 - n}}} \right)} \), \( \displaystyle \sum\limits_{n = 7}^\infty {\frac{{4 - n}}{{{n^2} + 1}}} \), \( \displaystyle \sum\limits_{n = 2}^\infty {\frac{{{{\left( { - 1} \right)}^{n - 3}}\left( {n + 2} \right)}}{{{5^{1 + 2n}}}}} \). % 5.3.2 Use the integral test to determine the convergence of a series. Free Practice Test Instructions: Choose your answer to the question and click 'Continue' to see how you did. 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 >> 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 << Ex 11.11.4 Show that \(\cos x\) is equal to its Taylor series for all \(x\) by showing that the limit of the error term is zero as N approaches infinity. Some infinite series converge to a finite value. 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 endobj /FontDescriptor 23 0 R /Widths[611.8 816 761.6 679.6 652.8 734 707.2 761.6 707.2 761.6 707.2 571.2 544 544 489.6 489.6 272 272 761.6 489.6 761.6 489.6 516.9 734 743.9 700.5 813 724.8 633.9 What is the 83rd term of the sequence 91, 87, 83, 79, ( = a. 1. 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. /LastChar 127 >> AP Calculus AB and BC: Chapter 9 -Infinite Sequences and Series : 9.4 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 Use the Comparison Test to determine whether each series in exercises 1 - 13 converges or diverges. L7s[AQmT*Z;HK%H0yqt1r8 Here are a set of practice problems for the Series and Sequences chapter of the Calculus II notes. If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. endobj 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. We will also give the Divergence Test for series in this section. )Ltgx?^eaT'&+n+hN4*D^UR!8UY@>LqS%@Cp/-12##DR}miBw6"ja+WjU${IH$5j!j-I1 /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 If the series is an alternating series, determine whether it converges absolutely, converges conditionally, or diverges. AP is a registered trademark of the College Board, which has not reviewed this resource. Sequences can be thought of as functions whose domain is the set of integers. 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). 2 6 points 2. Strip out the first 3 terms from the series \( \displaystyle \sum\limits_{n = 1}^\infty {\frac{{{2^{ - n}}}}{{{n^2} + 1}}} \). xTn0+,ITi](N@ fH2}W"UG'.% Z#>y{!9kJ+ When given a sum a[n], if the limit as n-->infinity does not exist or does not equal 0, the sum diverges. (answer), Ex 11.2.9 Compute \(\sum_{n=1}^\infty {3^n\over 5^{n+1}}\). Determine whether each series converges absolutely, converges conditionally, or diverges. We will also determine a sequence is bounded below, bounded above and/or bounded. We will focus on the basic terminology, limits of sequences and convergence of sequences in this section. To use integration by parts in Calculus, follow these steps: Decompose the entire integral (including dx) into two factors. It turns out the answer is no. /Type/Font Learn how this is possible, how we can tell whether a series converges, and how we can explore convergence in . Psychological Research & Experimental Design, All Teacher Certification Test Prep Courses. What if the interval is instead \([1,3/2]\)? << Choose your answer to the question and click 'Continue' to see how you did. 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. 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. Proofs for both tests are also given. A brick wall has 60 bricks in the first row, but each row has 3 fewer bricks than the previous one. Let the factor without dx equal u and the factor with dx equal dv. 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 Therefore the radius of convergence is R = , and the interval of convergence is ( - , ). /FirstChar 0 /FontDescriptor 20 0 R You may also use any of these materials for practice. << 762 689.7 1200.9 820.5 796.1 695.6 816.7 847.5 605.6 544.6 625.8 612.8 987.8 713.3 8 0 obj 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 Infinite sequences and series | AP/College Calculus BC - Khan Academy We also discuss differentiation and integration of power 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 stream /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 /Length 1247 endstream << Sequences & Series in Calculus Chapter Exam - Study.com /Subtype/Type1 Solution. /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 15 0 obj What is the sum of all the even integers from 2 to 250? 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 . S.QBt'(d|/"XH4!qbnEriHX)Gs2qN/G jq8$$< All other trademarks and copyrights are the property of their respective owners. 207 0 obj <> endobj 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. 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_. Here is a list of all the sections for which practice problems have been written as well as a brief description of the material covered in the notes for that particular section. Our mission is to provide a free, world-class education to anyone, anywhere. (answer), Ex 11.2.8 Compute \(\sum_{n=1}^\infty \left({3\over 5}\right)^n\). Which is the infinite sequence starting with 1 where each number is the previous number times 3? 5.3 The Divergence and Integral Tests - Calculus Volume 2 - OpenStax Alternating Series Test - In this section we will discuss using the Alternating Series Test to determine if an infinite series converges or diverges. %PDF-1.5 PDF Calc II: Practice Final Exam - Columbia University (answer), Ex 11.1.5 Determine whether \(\left\{{n+47\over\sqrt{n^2+3n}}\right\}_{n=1}^{\infty}\) converges or diverges. << 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 If a geometric series begins with the following term, what would the next term be? (answer), Ex 11.10.10 Use a combination of Maclaurin series and algebraic manipulation to find a series centered at zero for \( xe^{-x}\). Note that some sections will have more problems than others and some will have more or less of a variety of problems. When you have completed the free practice test, click 'View Results' to see your results. stream Alternating Series Test In this section we will discuss using the Alternating Series Test to determine if an infinite series converges or diverges. Math 1242: Calculus II - University of North Carolina at Charlotte 26 0 obj << /Type/Font Here are a set of practice problems for the Series and Sequences chapter of the Calculus II notes. Each review chapter is packed with equations, formulas, and examples with solutions, so you can study smarter and score a 5! Contact us by phone at (877)266-4919, or by mail at 100ViewStreet#202, MountainView, CA94041. << 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. 12 0 obj /Length 465 endobj We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. x[[o6~cX/e`ElRm'1%J$%v)tb]1U2sRV}.l%s\Y UD+q}O+J Learn how this is possible, how we can tell whether a series converges, and how we can explore convergence in Taylor and Maclaurin series. 979.2 489.6 489.6 489.6] << We will illustrate how we can find a series representation for indefinite integrals that cannot be evaluated by any other method. PDF Practice Problems Series & Sequences - MR. SOLIS' WEEBLY Convergence/Divergence of Series In this section we will discuss in greater detail the convergence and divergence of infinite series. Question 5 5. Then click 'Next Question' to answer the next question. 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 \(\sum_{n=1}^N {1\over n^4} + 0.005\). /BaseFont/BPHBTR+CMMI12 If you're seeing this message, it means we're having trouble loading external resources on our website. (answer), Ex 11.1.4 Determine whether \(\left\{{n^2+1\over (n+1)^2}\right\}_{n=0}^{\infty}\) converges or diverges. 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. 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 Donate or volunteer today! Good luck! (answer). 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 Good luck! We will examine Geometric Series, Telescoping Series, and Harmonic Series. Ex 11.10.8 Find the first four terms of the Maclaurin series for \(\tan x\) (up to and including the \( x^3\) term). /Subtype/Type1 Math 106 (Calculus II): old exams | Mathematics | Bates College The book contains eight practice tests five practice tests for Calculus AB and three practice tests for Calculus BC. /Type/Font In the previous section, we determined the convergence or divergence of several series by . Complementary General calculus exercises can be found for other Textmaps and can be accessed here. >> Learn for free about math, art, computer programming, economics, physics, chemistry, biology, medicine, finance, history, and more. What is the radius of convergence? If you're seeing this message, it means we're having trouble loading external resources on our website. 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. 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 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. Good luck! We also derive some well known formulas for Taylor series of \({\bf e}^{x}\) , \(\cos(x)\) and \(\sin(x)\) around \(x=0\). 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). (answer), Ex 11.9.2 Find a power series representation for \(1/(1-x)^2\). Khan Academy is a nonprofit with the mission of providing a free, world-class education for anyone, anywhere. In order to use either test the terms of the infinite series must be positive. /Filter /FlateDecode %PDF-1.5 % stream Math C185: Calculus II (Tran) 6: Sequences and Series 6.5: Comparison Tests 6.5E: Exercises for Comparison Test Expand/collapse global location 6.5E: Exercises for Comparison Test . For each of the following series, determine which convergence test is the best to use and explain why. /FontDescriptor 11 0 R OR sequences are lists of numbers, where the numbers may or may not be determined by a pattern. PDF Calculus II Series - Things to Consider - California State University Worksheets. /LastChar 127 /FirstChar 0 Most sections should have a range of difficulty levels in the problems although this will vary from section to section. Khan Academy is a 501(c)(3) nonprofit organization. Complementary General calculus exercises can be found for other Textmaps and can be accessed here. (1 point) Is the integral Z 1 1 1 x2 dx an improper integral? Which of the following is the 14th term of the sequence below? Integral Test In this section we will discuss using the Integral Test to determine if an infinite series converges or diverges. 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 endobj /Filter /FlateDecode Level up on all the skills in this unit and collect up to 2000 Mastery points! /Subtype/Type1 Maclaurin series of e, sin(x), and cos(x).
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