777.8 444.4 444.4 444.4 611.1 777.8 777.8 777.8 777.8] Ex 11.10.8 Find the first four terms of the Maclaurin series for \(\tan x\) (up to and including the \( x^3\) term). 1000 500 500 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 489.6 489.6 272 272 761.6 489.6 761.6 489.6 516.9 734 743.9 700.5 813 724.8 633.9 Choosing a Convergence Test | Calculus II - Lumen Learning xTn0+,ITi](N@ fH2}W"UG'.% Z#>y{!9kJ+ At this time, I do not offer pdfs for solutions to individual problems. endstream 666.7 1000 1000 1000 1000 1055.6 1055.6 1055.6 777.8 666.7 666.7 450 450 450 450 Sequences and Numerical series. /Length 1722 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). 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 Determine whether the following series converge or diverge. If a geometric series begins with the following term, what would the next term be? 611.8 897.2 734 761.6 666.2 761.6 720.6 544 707.2 734 734 1006 734 734 598.4 272 << Some infinite series converge to a finite value. /FirstChar 0 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@ " 207 0 obj <> endobj Strip out the first 3 terms from the series \( \displaystyle \sum\limits_{n = 1}^\infty {\frac{{{2^{ - n}}}}{{{n^2} + 1}}} \). copyright 2003-2023 Study.com. %PDF-1.5 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. /Name/F2 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 /LastChar 127 Calculus II - Series - The Basics (Practice Problems) - Lamar University Ex 11.7.9 Prove theorem 11.7.3, the root test. &/ r /Length 569 Which of the following sequences is NOT a geometric sequence? Choose your answer to the question and click 'Continue' to see how you did. When you have completed the free practice test, click 'View Results' to see your results. 12 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. xYKs6W(MCG:9iIO=(lkFRI$x$AMN/" J?~i~d cXf9o/r.&Lxy%/D-Yt+"LX]Sfp]Xl-aM_[6(*~mQbh*28AjZx0 =||. Quiz 1: 5 questions Practice what you've learned, and level up on the above skills. >> Series | Calculus 2 | Math | Khan Academy << Proofs for both tests are also given. 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. The steps are terms in the sequence. What is the 83rd term of the sequence 91, 87, 83, 79, ( = a. Then we can say that the series diverges without having to do any extra work. n a n converges if and only if the integral 1 f ( x) d x converges. A brick wall has 60 bricks in the first row, but each row has 3 fewer bricks than the previous one. You may also use any of these materials for practice. |: The Ratio Test shows us that regardless of the choice of x, the series converges. Integral Test: If a n = f ( n), where f ( x) is a non-negative non-increasing function, then. %|S#?\A@D-oS)lW=??nn}y]Tb!!o_=;]ha,J[. Alternating series test. 45 0 obj Sequences review (practice) | Series | Khan Academy (answer), Ex 11.2.3 Explain why \(\sum_{n=1}^\infty {3\over n}\) diverges. /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 A proof of the Ratio Test is also given. PDF Ap Calculus Ab Bc Kelley Copy - gny.salvationarmy.org 238 0 obj <>/Filter/FlateDecode/ID[<09CA7BCBAA751546BDEE3FEF56AF7BFA>]/Index[207 46]/Info 206 0 R/Length 137/Prev 582846/Root 208 0 R/Size 253/Type/XRef/W[1 3 1]>>stream Khan Academy is a 501(c)(3) nonprofit organization. Ex 11.7.5 \(\sum_{n=0}^\infty (-1)^{n}{3^n\over 5^n}\) (answer), Ex 11.7.6 \(\sum_{n=1}^\infty {n!\over n^n}\) (answer), Ex 11.7.7 \(\sum_{n=1}^\infty {n^5\over n^n}\) (answer), Ex 11.7.8 \(\sum_{n=1}^\infty {(n! Our mission is to provide a free, world-class education to anyone, anywhere. 70 terms. bmkraft7. We will also see how we can use the first few terms of a power series to approximate a function. Ex 11.7.4 Compute \(\lim_{n\to\infty} |a_n|^{1/n}\) for the series \(\sum 1/n\). 21 terms. (answer), Ex 11.9.2 Find a power series representation for \(1/(1-x)^2\). 979.2 489.6 489.6 489.6] 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 n = 1 n2 + 2n n3 + 3n2 + 1. Good luck! For each function, find the Maclaurin series or Taylor series centered at $a$, and the radius of convergence. /Type/Font 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. Below are some general cases in which each test may help: P-Series Test: The series be written in the form: P 1 np Geometric Series Test: When the series can be written in the form: P a nrn1 or P a nrn Direct Comparison Test: When the given series, a 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? endobj Given that \( \displaystyle \sum\limits_{n = 0}^\infty {\frac{1}{{{n^3} + 1}}} = 1.6865\) determine the value of \( \displaystyle \sum\limits_{n = 2}^\infty {\frac{1}{{{n^3} + 1}}} \). Integral test. /Subtype/Type1 Here are a set of practice problems for the Series and Sequences chapter of the Calculus II notes. 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 ,vEmO8/OuNVRaLPqB.*l. /FirstChar 0 The following is a list of worksheets and other materials related to Math 129 at the UA. >> 5.3.1 Use the divergence test to determine whether a series converges or diverges. PDF FINAL EXAM CALCULUS 2 - Department of Mathematics (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 \(\sum_{n=1}^N {\ln n\over n^2} + 0.005\). 11.E: Sequences and Series (Exercises) These are homework exercises to accompany David Guichard's "General Calculus" Textmap. Determine whether the sequence converges or diverges. %PDF-1.5 % Choose your answer to the question and click 'Continue' to see how you did. In other words, a series is the sum of a sequence. endstream Alternating Series Test - In this section we will discuss using the Alternating Series Test to determine if an infinite series converges or diverges. Each term is the difference of the previous two terms. Ex 11.1.3 Determine whether \(\{\sqrt{n+47}-\sqrt{n}\}_{n=0}^{\infty}\) converges or diverges. 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 11.10.10 Use a combination of Maclaurin series and algebraic manipulation to find a series centered at zero for \( xe^{-x}\). endstream endobj 208 0 obj <. >> 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. Ex 11.7.3 Compute \(\lim_{n\to\infty} |a_n|^{1/n}\) for the series \(\sum 1/n^2\). /Subtype/Type1 If L = 1, then the test is inconclusive. 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. /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 /Subtype/Type1 /Filter[/FlateDecode] At this time, I do not offer pdf's for solutions to individual problems. Calculus 2. Note that some sections will have more problems than others and some will have more or less of a variety of problems. >> nn = 0. Which of the following is the 14th term of the sequence below? 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 /BaseFont/BPHBTR+CMMI12 PDF Arithmetic Sequences And Series Practice Problems Sequences & Series in Calculus Chapter Exam - Study.com /LastChar 127 If it con-verges, nd the limit. The book contains eight practice tests five practice tests for Calculus AB and three practice tests for Calculus BC. )^2\over n^n}\) (answer). Determine whether each series converges or diverges. Taylor Series In this section we will discuss how to find the Taylor/Maclaurin Series for a function. Therefore the radius of convergence is R = , and the interval of convergence is ( - , ). Convergence/Divergence of Series In this section we will discuss in greater detail the convergence and divergence of infinite series. Remark. Series are sums of multiple terms. 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 15 0 obj Here are a set of practice problems for the Series and Sequences chapter of the Calculus II notes. Donate or volunteer today! }\) (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! (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. Divergence Test. 252 0 obj <>stream 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 When you have completed the free practice test, click 'View Results' to see your results. 24 0 obj stream Chapter 10 : Series and Sequences. % L7s[AQmT*Z;HK%H0yqt1r8 (You may want to use Sage or a similar aid.) (answer), Ex 11.2.8 Compute \(\sum_{n=1}^\infty \left({3\over 5}\right)^n\). /FontDescriptor 20 0 R The chapter headings refer to Calculus, Sixth Edition by Hughes-Hallett et al. Remark. 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. Learn how this is possible, how we can tell whether a series converges, and how we can explore convergence in . /Type/Font 611.8 897.2 734 761.6 666.2 761.6 720.6 544 707.2 734 734 1006 734 734 598.4 272 Consider the series n a n. Divergence Test: If lim n a n 0, then n a n diverges. 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_. copyright 2003-2023 Study.com. 17 0 obj MATH 126 Medians and Such. This page titled 11.E: Sequences and Series (Exercises) is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by David Guichard. 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 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 >> Some infinite series converge to a finite value. 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 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. 1. >> Choose the equation below that represents the rule for the nth term of the following geometric sequence: 128, 64, 32, 16, 8, . /Name/F1 << /BaseFont/UNJAYZ+CMR12 Which of the following sequences follows this formula. /Name/F6 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 . Maclaurin series of e, sin(x), and cos(x). 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 /Length 465 It turns out the answer is no. Math 106 (Calculus II): old exams | Mathematics | Bates College Which of the following sequences is NOT a geometric sequence? We will illustrate how we can find a series representation for indefinite integrals that cannot be evaluated by any other method. 805.6 805.6 1277.8 1277.8 811.1 811.1 875 875 666.7 666.7 666.7 666.7 666.7 666.7 665 570.8 924.4 812.6 568.1 670.2 380.8 380.8 380.8 979.2 979.2 410.9 514 416.3 421.4 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, 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, Interval of convergence for derivative and integral, Integrals & derivatives of functions with known power series, 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. /Length 2492 (answer), Ex 11.2.4 Compute \(\sum_{n=0}^\infty {4\over (-3)^n}- {3\over 3^n}\). Determine whether each series converges absolutely, converges conditionally, or diverges. The Alternating Series Test can be used only if the terms of the series alternate in sign. S.QBt'(d|/"XH4!qbnEriHX)Gs2qN/G jq8$$< /BaseFont/CQGOFL+CMSY10 PDF Practice Problems Series & Sequences - MR. SOLIS' WEEBLY (answer). 531.3 531.3 531.3] 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.