calculus 2 series and sequences practice test

When you have completed the free practice test, click 'View Results' to see your results. endobj 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 Math 106 (Calculus II): old exams. Which of the following sequences is NOT a geometric sequence? /Type/Font Root Test In this section we will discuss using the Root Test to determine if an infinite series converges absolutely or diverges. /LastChar 127 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. /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 /Name/F1 When given a sum a[n], if the limit as n-->infinity does not exist or does not equal 0, the sum diverges. 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. 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 (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 \(\sum_{n=1}^N {\ln n\over n^2} + 0.005\). 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. All other trademarks and copyrights are the property of their respective owners. 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 Psychological Research & Experimental Design, All Teacher Certification Test Prep Courses. \ _* %l~G"tytO(J*l+X@ uE: m/ ~&Q24Nss(7F!ky=4 Mijo8t;v A proof of the Integral Test is also given. Which of the following sequences follows this formula? << Choose your answer to the question and click 'Continue' to see how you did. 5.3.1 Use the divergence test to determine whether a series converges or diverges. xYKs6W(MCG:9iIO=(lkFRI$x$AMN/" J?~i~d cXf9o/r.&Lxy%/D-Yt+"LX]Sfp]Xl-aM_[6(*~mQbh*28AjZx0 =||. !A1axw)}p]WgxmkFftu |: The Ratio Test shows us that regardless of the choice of x, the series converges. The Alternating Series Test can be used only if the terms of the 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. /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 (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 \(\sum_{n=0}^N {1\over e^n} + 10^{-4}\). If it con-verges, nd the limit. 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] Images. Comparison tests. Ratio test. 252 0 obj <>stream /BaseFont/SFGTRF+CMSL12 n = 1 n 2 + 2 n n 3 + 3 n . << << /LastChar 127 Determine whether each series converges absolutely, converges conditionally, or diverges. 12 0 obj Indiana Core Assessments Mathematics: Test Prep & Study Guide. 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). (answer). (answer), Ex 11.2.6 Compute \(\sum_{n=0}^\infty {4^{n+1}\over 5^n}\). To log in and use all the features of Khan Academy, please enable JavaScript in your browser. Each term is the sum of the previous two terms. Math 129 - Calculus II. Which equation below represents a geometric sequence? 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 /FontDescriptor 11 0 R If you're seeing this message, it means we're having trouble loading external resources on our website. Here are a set of practice problems for the Series and Sequences chapter of the Calculus II notes. Which of the following sequences follows this formula. endstream endobj 208 0 obj <. Then click 'Next Question' to answer the next question. AP is a registered trademark of the College Board, which has not reviewed this resource. 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 979.2 489.6 489.6 489.6] (answer), Ex 11.9.2 Find a power series representation for \(1/(1-x)^2\). 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, Convergent & divergent geometric series (with manipulation), 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, Integrals & derivatives of functions with known power series, Interval of convergence for derivative and integral, Converting explicit series terms to summation notation, Converting explicit series terms to summation notation (n 2), 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. /LastChar 127 Section 10.3 : Series - Basics. /Length 569 (b) Note that some sections will have more problems than others and some will have more or less of a variety of problems. OR sequences are lists of numbers, where the numbers may or may not be determined by a pattern. 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. Level up on all the skills in this unit and collect up to 2000 Mastery points! 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 For problems 1 - 3 perform an index shift so that the series starts at n = 3 n = 3. 18 0 obj Strategy for Series In this section we give a general set of guidelines for determining which test to use in determining if an infinite series will converge or diverge. Chapter 10 : Series and Sequences. % 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. hbbd```b``~"A$" "Y`L6`RL,-`sA$w64= f[" RLMu;@jAl[`3H^Ne`?$4 883.8 992.6 761.6 272 272 489.6 816 489.6 816 761.6 272 380.8 380.8 489.6 761.6 272 The chapter headings refer to Calculus, Sixth Edition by Hughes-Hallett et al. Infinite series are sums of an infinite number of terms. 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. 888.9 888.9 888.9 888.9 888.9 888.9 888.9 666.7 875 875 875 875 611.1 611.1 833.3 %PDF-1.2 Comparison Test: This applies . Learning Objectives. /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 Here are a set of practice problems for the Series and Sequences chapter of the Calculus II notes. /LastChar 127 stream When you have completed the free practice test, click 'View Results' to see your results. Choose the equation below that represents the rule for the nth term of the following geometric sequence: 128, 64, 32, 16, 8, . 1 2 + 1 4 + 1 8 + = n=1 1 2n = 1 We will need to be careful, but it turns out that we can . 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. 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 If you . Example 1. Accessibility StatementFor more information contact us atinfo@libretexts.org. All other trademarks and copyrights are the property of their respective owners. endobj Here are a set of practice problems for the Series and Sequences chapter of the Calculus II notes. Calculus II-Sequences and Series. 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. 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 (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. Premium members get access to this practice exam along with our entire library of lessons taught by subject matter experts. << /Subtype/Type1 (answer), Ex 11.1.4 Determine whether \(\left\{{n^2+1\over (n+1)^2}\right\}_{n=0}^{\infty}\) converges or diverges. /BaseFont/CQGOFL+CMSY10 Choose your answer to the question and click 'Continue' to see how you did. Find the radius and interval of convergence for each series. 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. 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. Some infinite series converge to a finite value. We will illustrate how we can find a series representation for indefinite integrals that cannot be evaluated by any other method. 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. >> endstream endobj startxref 62 0 obj endstream /Subtype/Type1 /Type/Font (answer), Ex 11.3.12 Find an \(N\) so that \(\sum_{n=2}^\infty {1\over n(\ln n)^2}\) is between \(\sum_{n=2}^N {1\over n(\ln n)^2}\) and \(\sum_{n=2}^N {1\over n(\ln n)^2} + 0.005\). 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 1062.5 556.5 425.2 527.8 579.5 613.4 636.6 609.7 458.2 577.1 808.9 505 354.2 641.4 979.2 8 0 obj Series The Basics In this section we will formally define an infinite series. stream 979.2 489.6 489.6 489.6] >> /Filter /FlateDecode 816 816 272 299.2 489.6 489.6 489.6 489.6 489.6 734 435.2 489.6 707.2 761.6 489.6 These are homework exercises to accompany David Guichard's "General Calculus" Textmap. (answer). Choose your answer to the question and click 'Continue' to see how you did. << (answer), Ex 11.1.6 Determine whether \(\left\{{2^n\over n! % Therefore the radius of convergence is R = , and the interval of convergence is ( - , ). 5.3.2 Use the integral test to determine the convergence of a series. 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 /Type/Font (answer), Ex 11.9.3 Find a power series representation for \( 2/(1-x)^3\). If a geometric series begins with the following term, what would the next term be? 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 Ex 11.1.2 Use the squeeze theorem to show that \(\lim_{n\to\infty} {n!\over n^n}=0\). 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 . What is the sum of all the even integers from 2 to 250? Ex 11.1.3 Determine whether \(\{\sqrt{n+47}-\sqrt{n}\}_{n=0}^{\infty}\) converges or diverges. (answer). /Type/Font /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 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. The numbers used come from a sequence. 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. Don't all infinite series grow to infinity? %|S#?\A@D-oS)lW=??nn}y]Tb!!o_=;]ha,J[. Which of the following sequences is NOT a geometric sequence? If it converges, compute the limit. Learn how this is possible, how we can tell whether a series converges, and how we can explore convergence in Taylor and Maclaurin series. /Length 2492 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 Calculus (single and multi-variable) Ordinary Differential equations (upto 2nd order linear with Laplace transforms, including Dirac Delta functions and Fourier Series. Series are sums of multiple terms. 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. Research Methods Midterm. 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. /Name/F2 Ex 11.4.1 \(\sum_{n=1}^\infty {(-1)^{n-1}\over 2n+5}\) (answer), Ex 11.4.2 \(\sum_{n=4}^\infty {(-1)^{n-1}\over \sqrt{n-3}}\) (answer), Ex 11.4.3 \(\sum_{n=1}^\infty (-1)^{n-1}{n\over 3n-2}\) (answer), Ex 11.4.4 \(\sum_{n=1}^\infty (-1)^{n-1}{\ln n\over n}\) (answer), Ex 11.4.5 Approximate \(\sum_{n=1}^\infty (-1)^{n-1}{1\over n^3}\) to two decimal places. Your instructor might use some of these in class. 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. SAT Practice Questions- All Maths; SAT Practice Test Questions- Reading , Writing and Language; KS 1-2 Math, Science and SAT . 2 6 points 2. Alternating Series Test - In this section we will discuss using the Alternating Series Test to determine if an infinite series converges or diverges. All rights reserved. Remark. Let the factor without dx equal u and the factor with dx equal dv. Other sets by this creator. 531.3 531.3 531.3] 611.8 897.2 734 761.6 666.2 761.6 720.6 544 707.2 734 734 1006 734 734 598.4 272 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 S.QBt'(d|/"XH4!qbnEriHX)Gs2qN/G jq8$$< endobj Ex 11.1.3 Determine whether {n + 47 n} . (answer), Ex 11.2.1 Explain why \(\sum_{n=1}^\infty {n^2\over 2n^2+1}\) diverges. Quiz 2: 8 questions Practice what you've learned, and level up on the above skills. >> << stream xWKoFWlojCpP NDED$(lq"g|3g6X_&F1BXIM5d gOwaN9c,r|9 xTn0+,ITi](N@ fH2}W"UG'.% Z#>y{!9kJ+ 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 /FirstChar 0 >> If it converges, compute the limit. Solution. Sequences and Numerical 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 We will focus on the basic terminology, limits of sequences and convergence of sequences in this section. Sequences can be thought of as functions whose domain is the set of integers. (answer), Ex 11.1.5 Determine whether \(\left\{{n+47\over\sqrt{n^2+3n}}\right\}_{n=1}^{\infty}\) converges or diverges. /Subtype/Type1 (answer). 772.4 811.3 431.9 541.2 833 666.2 947.3 784.1 748.3 631.1 775.5 745.3 602.2 573.9 All rights reserved. Our mission is to provide a free, world-class education to anyone, anywhere. Ex 11.7.4 Compute \(\lim_{n\to\infty} |a_n|^{1/n}\) for the series \(\sum 1/n\). 15 0 obj Complementary General calculus exercises can be found for other Textmaps and can be accessed here. Quiz 1: 5 questions Practice what you've learned, and level up on the above skills. Sequences and Series. 1. 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. In other words, a series is the sum of a sequence. << A proof of the Root Test is also given. )^2\over n^n}\) (answer). 4 avwo/MpLv) _C>5p*)i=^m7eE. (You may want to use Sage or a similar aid.) >> in calculus coursesincluding Calculus, Calculus II, Calculus III, AP Calculus and Precalculus. 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 Applications of Series In this section we will take a quick look at a couple of applications of series. At this time, I do not offer pdf's for solutions to individual problems. endobj 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! The sum of the steps forms an innite series, the topic of Section 10.2 and the rest of Chapter 10. (answer), Ex 11.2.7 Compute \(\sum_{n=0}^\infty {3^{n+1}\over 7^{n+1}}\). 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 26 0 obj (answer), Ex 11.2.2 Explain why \(\sum_{n=1}^\infty {5\over 2^{1/n}+14}\) diverges. >> endstream In the previous section, we determined the convergence or divergence of several series by . Ex 11.10.8 Find the first four terms of the Maclaurin series for \(\tan x\) (up to and including the \( x^3\) term). We will illustrate how partial sums are used to determine if an infinite series converges or diverges. Ex 11.7.1 Compute \(\lim_{n\to\infty} |a_{n+1}/a_n|\) for the series \(\sum 1/n^2\). At this time, I do not offer pdfs for solutions to individual problems. >> 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. Alternating Series Test In this section we will discuss using the Alternating Series Test to determine if an infinite 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 . 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. 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. Taylor Series In this section we will discuss how to find the Taylor/Maclaurin Series for a function. (answer), Ex 11.2.8 Compute \(\sum_{n=1}^\infty \left({3\over 5}\right)^n\). bmkraft7. /BaseFont/VMQJJE+CMR8 L7s[AQmT*Z;HK%H0yqt1r8 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. Then click 'Next Question' to answer the next question. /Widths[606.7 816 748.3 679.6 728.7 811.3 765.8 571.2 652.8 598 757.6 622.8 552.8 413.2 531.3 826.4 295.1 354.2 295.1 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 /FontDescriptor 14 0 R &/ r 21 terms. 17 0 obj >> >> 883.8 992.6 761.6 272 272 489.6 816 489.6 816 761.6 272 380.8 380.8 489.6 761.6 272 Proofs for both tests are also given. /Length 1722 Strip out the first 3 terms from the series \( \displaystyle \sum\limits_{n = 1}^\infty {\frac{{{2^{ - n}}}}{{{n^2} + 1}}} \). Good luck! We will also give the Divergence Test for series in this section. 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 Then we can say that the series diverges without having to do any extra work. MULTIPLE CHOICE: Circle the best answer. You may also use any of these materials for practice. (1 point) Is the integral Z 1 1 1 x2 dx an improper integral? Choose your answer to the question and click 'Continue' to see how you did. 2.(a). /Name/F5 Good luck! (answer). To log in and use all the features of Khan Academy, please enable JavaScript in your browser. (answer). Bottom line -- series are just a lot of numbers added together. /Subtype/Type1 Ex 11.9.5 Find a power series representation for \(\int\ln(1-x)\,dx\). If it converges, compute the limit. (a) $\sum_{n=1}^{\infty} \frac{(-1)^n}{\sqrt{n}}$ (b) $\sum_{n=1}^{\infty}(-1)^n \frac{n}{2 n-1}$ stream We will also determine a sequence is bounded below, bounded above and/or bounded. 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. Then determine if the series converges or diverges. Which is the infinite sequence starting with 1 where each number is the previous number times 3? Then click 'Next Question' to answer the . /Filter /FlateDecode /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@ " 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 Which rule represents the nth term in the sequence 9, 16, 23, 30? 21 0 obj Which of the sequences below has the recursive rule where each number is the previous number times 2? Most sections should have a range of difficulty levels in the problems although this will vary from section to section. 611.8 897.2 734 761.6 666.2 761.6 720.6 544 707.2 734 734 1006 734 734 598.4 272 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. 68 0 obj copyright 2003-2023 Study.com. ZrNRG{I~(iw%0W5b)8*^ yyCCy~Cg{C&BPsTxp%p A brick wall has 60 bricks in the first row, but each row has 3 fewer bricks than the previous one. /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 >> /Filter /FlateDecode 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 (answer). /Filter /FlateDecode x=S0 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? (answer), Ex 11.10.9 Use a combination of Maclaurin series and algebraic manipulation to find a series centered at zero for \( x\cos (x^2)\). endobj }\right\}_{n=0}^{\infty}\) converges or diverges. << stream 31 terms. 1 2, 1 4, 1 8, Sequences of values of this type is the topic of this rst section. /BaseFont/BPHBTR+CMMI12 A proof of the Alternating Series Test is also given. 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 Premium members get access to this practice exam along with our entire library of lessons taught by subject matter experts. << Then click 'Next Question' to answer the next question. 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.
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