After each bounce, the ball reaches a height that is 2/3 of the height from which it previously fell. << 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 xTn0+,ITi](N@ fH2}W"UG'.% Z#>y{!9kJ+ 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. |: The Ratio Test shows us that regardless of the choice of x, the series converges. For each function, find the Maclaurin series or Taylor series centered at $a$, and the radius of convergence. Calculus II - Sequences and Series Flashcards | Quizlet Images. << What if the interval is instead \([1,3/2]\)? }\) (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! Math 129 - Calculus II. Determine whether each series converges or diverges. ]^e-V!2 F. 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. Learning Objectives. Calculus II - Series - The Basics (Practice Problems) - Lamar University Which of the following is the 14th term of the sequence below? Khan Academy is a 501(c)(3) nonprofit organization. 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. ZrNRG{I~(iw%0W5b)8*^ yyCCy~Cg{C&BPsTxp%p Sequences & Series in Calculus Chapter Exam - Study.com (answer). /LastChar 127 Each term is the difference of the previous two terms. % (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 \(\sum_{n=0}^N {1\over e^n} + 10^{-4}\). /FirstChar 0 Derivatives, Integrals, Sequences & Series, and Vector Valued Functions. Donate or volunteer today! Integral Test In this section we will discuss using the Integral Test to determine if an infinite series converges or diverges. copyright 2003-2023 Study.com. (answer), Ex 11.4.6 Approximate \(\sum_{n=1}^\infty (-1)^{n-1}{1\over n^4}\) to two decimal places. (answer). /Subtype/Type1 Martha_Austin Teacher. The Root 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. To use integration by parts in Calculus, follow these steps: Decompose the entire integral (including dx) into two factors. Each review chapter is packed with equations, formulas, and examples with solutions, so you can study smarter and score a 5! We will also see how we can use the first few terms of a power series to approximate a function. 18 0 obj (answer). If it converges, compute the limit. /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 21 0 obj 24 0 obj Good luck! 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 Ex 11.1.2 Use the squeeze theorem to show that \(\lim_{n\to\infty} {n!\over n^n}=0\). OR sequences are lists of numbers, where the numbers may or may not be determined by a pattern. For each of the following series, determine which convergence test is the best to use and explain why. 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 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. 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. Which one of these sequences is a finite sequence? /Filter[/FlateDecode] endobj 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 Contact us by phone at (877)266-4919, or by mail at 100ViewStreet#202, MountainView, CA94041. 21 terms. /FontDescriptor 14 0 R /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 ,vEmO8/OuNVRaLPqB.*l. Learn for free about math, art, computer programming, economics, physics, chemistry, biology, medicine, finance, history, and more. n = 1 n 2 + 2 n n 3 + 3 n . Maclaurin series of e, sin(x), and cos(x). Sequences and Numerical series. All other trademarks and copyrights are the property of their respective owners. Series | Calculus 2 | Math | Khan Academy If the series is an alternating series, determine whether it converges absolutely, converges conditionally, or diverges. )^2\over n^n}\) (answer). Proofs for both tests are also given. endobj Infinite series are sums of an infinite number of terms. Indiana Core Assessments Mathematics: Test Prep & Study Guide. Psychological Research & Experimental Design, All Teacher Certification Test Prep Courses. Determine whether the following series converge or diverge. My calculus 2 exam on sequence, infinite series & power seriesThe exam: https://bit.ly/36OHYcsAll the convergence tests: https://bit.ly/2IzqokhBest friend an. Alternating series test. /Filter /FlateDecode Psychological Research & Experimental Design, All Teacher Certification Test Prep Courses. (answer), Ex 11.1.4 Determine whether \(\left\{{n^2+1\over (n+1)^2}\right\}_{n=0}^{\infty}\) converges or diverges. /Length 2492 >> Determine whether the series converge or diverge. Some infinite series converge to a finite value. To log in and use all the features of Khan Academy, please enable JavaScript in your browser. Choose your answer to the question and click 'Continue' to see how you did. Calculus II-Sequences and Series. We will also give many of the basic facts, properties and ways we can use to manipulate a series. Comparison Test: This applies . When you have completed the free practice test, click 'View Results' to see your results. 5.3.2 Use the integral test to determine the convergence of a series. A proof of the Alternating Series Test is also given. >> endobj 5.3.3 Estimate the value of a series by finding bounds on its remainder term. 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. %PDF-1.2 Good luck! Bottom line -- series are just a lot of numbers added together. (answer). endstream If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. Course summary; . (a) $\sum_{n=1}^{\infty} \frac{(-1)^n}{\sqrt{n}}$ (b) $\sum_{n=1}^{\infty}(-1)^n \frac{n}{2 n-1}$ Calculus II - Series & Sequences (Practice Problems) - Lamar University /BaseFont/CQGOFL+CMSY10 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 Khan Academy is a nonprofit with the mission of providing a free, world-class education for anyone, anywhere. endobj bmkraft7. /Filter /FlateDecode 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 Series The Basics In this section we will formally define an infinite series. endstream Consider the series n a n. Divergence Test: If lim n a n 0, then n a n diverges. Some infinite series converge to a finite value. 666.7 1000 1000 1000 1000 1055.6 1055.6 1055.6 777.8 666.7 666.7 450 450 450 450 /Length 465 15 0 obj 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). Which is the infinite sequence starting with 1 where each number is the previous number times 3? << Quiz 1: 5 questions Practice what you've learned, and level up on the above skills. SAT Practice Questions- All Maths; SAT Practice Test Questions- Reading , Writing and Language; KS 1-2 Math, Science and SAT . It turns out the answer is no. Integral test. What is the sum of all the even integers from 2 to 250? 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 The steps are terms in the sequence. Question 5 5. 8 0 obj The book contains eight practice tests five practice tests for Calculus AB and three practice tests for Calculus BC. >> << PDF FINAL EXAM CALCULUS 2 - Department of Mathematics 11.E: Sequences and Series (Exercises) These are homework exercises to accompany David Guichard's "General Calculus" Textmap. Chapters include Linear 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 . 979.2 489.6 489.6 489.6] Infinite series are sums of an infinite number of terms. We will determine if a sequence in an increasing sequence or a decreasing sequence and hence if it is a monotonic sequence. Comparison tests. /BaseFont/BPHBTR+CMMI12 PDF Practice Problems Series & Sequences - MR. SOLIS' WEEBLY 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. A proof of the Root Test is also given. The chapter headings refer to Calculus, Sixth Edition by Hughes-Hallett et al. hb```9B 7N0$K3 }M[&=cx`c$Y&a YG&lwG=YZ}w{l;r9P"J,Zr]Ngc E4OY%8-|\C\lVn@`^) E 3iL`h`` !f s9B`)qLa0$FQLN$"H&8001a2e*9y,Xs~z1111)QSEJU^|2n[\\5ww0EHauC8Gt%Y>2@ " 777.8 444.4 444.4 444.4 611.1 777.8 777.8 777.8 777.8] (answer), Ex 11.9.4 Find a power series representation for \( 1/(1-x)^3\). Published by Wiley. (answer), Ex 11.2.1 Explain why \(\sum_{n=1}^\infty {n^2\over 2n^2+1}\) diverges. More on Sequences In this section we will continue examining sequences. 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 PDF Read Free Answers To Algebra 2 Practice B Ellipses All other trademarks and copyrights are the property of their respective owners.