>> 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. The sum of the steps forms an innite series, the topic of Section 10.2 and the rest of Chapter 10. We will also give the Divergence Test for series in this section. 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 /Name/F1 If it converges, compute the limit. 68 0 obj /Subtype/Type1 We will determine if a sequence in an increasing sequence or a decreasing sequence and hence if it is a monotonic sequence. 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 Good luck! << Find the radius and interval of convergence for each series. To use the Geometric Series formula, the function must be able to be put into a specific form, which is often impossible. (answer), Ex 11.4.6 Approximate \(\sum_{n=1}^\infty (-1)^{n-1}{1\over n^4}\) to two decimal places. 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@ " 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 If a geometric series begins with the following term, what would the next term be? Note that some sections will have more problems than others and some will have more or less of a variety of problems. Complementary General calculus exercises can be found for other Textmaps and can be accessed here. % Ratio Test In this section we will discuss using the Ratio Test to determine if an infinite series converges absolutely or diverges. YesNo 2.(b). Each term is the product of the two previous terms. We will illustrate how partial sums are used to determine if an infinite series converges or diverges. 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 (answer), Ex 11.2.3 Explain why \(\sum_{n=1}^\infty {3\over n}\) diverges. (1 point) Is the integral Z 1 1 1 x2 dx an improper integral? n a n converges if and only if the integral 1 f ( x) d x converges. (answer). << >> 207 0 obj <> endobj 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. A brick wall has 60 bricks in the first row, but each row has 3 fewer bricks than the previous one. When given a sum a[n], if the limit as n-->infinity does not exist or does not equal 0, the sum diverges. 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. 5.3.3 Estimate the value of a series by finding bounds on its remainder term. 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. If L = 1, then the test is inconclusive. 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 Other sets by this creator. 21 0 obj Given that \( \displaystyle \sum\limits_{n = 0}^\infty {\frac{1}{{{n^3} + 1}}} = 1.6865\) determine the value of \( \displaystyle \sum\limits_{n = 2}^\infty {\frac{1}{{{n^3} + 1}}} \). 777.8 444.4 444.4 444.4 611.1 777.8 777.8 777.8 777.8] Ex 11.8.1 \(\sum_{n=0}^\infty n x^n\) (answer), Ex 11.8.2 \(\sum_{n=0}^\infty {x^n\over n! At this time, I do not offer pdf's for . These are homework exercises to accompany David Guichard's "General Calculus" Textmap. Absolute and conditional convergence. Quiz 1: 5 questions Practice what you've learned, and level up on the above skills. We also discuss differentiation and integration of power series. /Length 1722 /LastChar 127 /Length 569 Which one of these sequences is a finite sequence? 24 0 obj Khan Academy is a nonprofit with the mission of providing a free, world-class education for anyone, anywhere. 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 Maclaurin series of e, sin(x), and cos(x). nn = 0. 9 0 obj MULTIPLE CHOICE: Circle the best answer. endstream endobj 208 0 obj <. << 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 /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 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. Learning Objectives. Here are a set of practice problems for the Series and Sequences chapter of the Calculus II notes. (a) $\sum_{n=1}^{\infty} \frac{(-1)^n}{\sqrt{n}}$ (b) $\sum_{n=1}^{\infty}(-1)^n \frac{n}{2 n-1}$ stream Then click 'Next Question' to answer the . Remark. Therefore the radius of convergence is R = , and the interval of convergence is ( - , ). In order to use either test the terms of the infinite series must be positive. Calculus (single and multi-variable) Ordinary Differential equations (upto 2nd order linear with Laplace transforms, including Dirac Delta functions and Fourier Series. After each bounce, the ball reaches a height that is 2/3 of the height from which it previously fell. 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 At this time, I do not offer pdf's for solutions to individual problems. (answer), Ex 11.2.1 Explain why \(\sum_{n=1}^\infty {n^2\over 2n^2+1}\) diverges. /FontDescriptor 14 0 R Khan Academy is a 501(c)(3) nonprofit organization. If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. Ex 11.6.1 \(\sum_{n=1}^\infty (-1)^{n-1}{1\over 2n^2+3n+5}\) (answer), Ex 11.6.2 \(\sum_{n=1}^\infty (-1)^{n-1}{3n^2+4\over 2n^2+3n+5}\) (answer), Ex 11.6.3 \(\sum_{n=1}^\infty (-1)^{n-1}{\ln n\over n}\) (answer), Ex 11.6.4 \(\sum_{n=1}^\infty (-1)^{n-1} {\ln n\over n^3}\) (answer), Ex 11.6.5 \(\sum_{n=2}^\infty (-1)^n{1\over \ln n}\) (answer), Ex 11.6.6 \(\sum_{n=0}^\infty (-1)^{n} {3^n\over 2^n+5^n}\) (answer), Ex 11.6.7 \(\sum_{n=0}^\infty (-1)^{n} {3^n\over 2^n+3^n}\) (answer), Ex 11.6.8 \(\sum_{n=1}^\infty (-1)^{n-1} {\arctan n\over n}\) (answer). In the previous section, we determined the convergence or divergence of several series by . 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 Calculus 2. 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. We will examine Geometric Series, Telescoping Series, and Harmonic Series. %|S#?\A@D-oS)lW=??nn}y]Tb!!o_=;]ha,J[. A proof of the Integral Test is also given. Donate or volunteer today! (answer), Ex 11.2.4 Compute \(\sum_{n=0}^\infty {4\over (-3)^n}- {3\over 3^n}\). What is the 83rd term of the sequence 91, 87, 83, 79, ( = a. %PDF-1.2 Alternating series test. 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 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. Premium members get access to this practice exam along with our entire library of lessons taught by subject matter experts. If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. L7s[AQmT*Z;HK%H0yqt1r8 Determine whether each series converges or diverges. Psychological Research & Experimental Design, All Teacher Certification Test Prep Courses. We will also give many of the basic facts, properties and ways we can use to manipulate a series. Comparison Test: This applies . 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. Part II. 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. 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 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. Alternating Series Test In this section we will discuss using the Alternating Series Test to determine if an infinite series converges or diverges. Determine whether the sequence converges or diverges. 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. n = 1 n 2 + 2 n n 3 + 3 n . 2.(a). 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. (answer), Ex 11.1.4 Determine whether \(\left\{{n^2+1\over (n+1)^2}\right\}_{n=0}^{\infty}\) converges or diverges. endobj %PDF-1.5 % << /Name/F6 Good luck! stream Determine whether the series is convergent or divergent. The chapter headings refer to Calculus, Sixth Edition by Hughes-Hallett et al. /Length 2492 (answer), Ex 11.10.10 Use a combination of Maclaurin series and algebraic manipulation to find a series centered at zero for \( xe^{-x}\). Which is the infinite sequence starting with 1 where each number is the previous number times 3? xYKs6W(MCG:9iIO=(lkFRI$x$AMN/" J?~i~d cXf9o/r.&Lxy%/D-Yt+"LX]Sfp]Xl-aM_[6(*~mQbh*28AjZx0 =||. If the series is an alternating series, determine whether it converges absolutely, converges conditionally, or diverges. 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 /Type/Font Ratio test. Ex 11.1.1 Compute \(\lim_{x\to\infty} x^{1/x}\). 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. All rights reserved. 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. %%EOF When you have completed the free practice test, click 'View Results' to see your results. 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. /FontDescriptor 23 0 R 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 If you . endobj /Type/Font /Filter /FlateDecode Each review chapter is packed with equations, formulas, and examples with solutions, so you can study smarter and score a 5! (answer), Ex 11.1.5 Determine whether \(\left\{{n+47\over\sqrt{n^2+3n}}\right\}_{n=1}^{\infty}\) converges or diverges. /Length 465 For each function, find the Maclaurin series or Taylor series centered at $a$, and the radius of convergence. Integral test. /Type/Font Math 129 - Calculus II. We will also determine a sequence is bounded below, bounded above and/or bounded. 252 0 obj <>stream &/ r Defining convergent and divergent infinite series, Determining absolute or conditional convergence, Finding Taylor polynomial approximations of functions, Radius and interval of convergence of power series, Finding Taylor or Maclaurin series for a function. More on Sequences In this section we will continue examining sequences. Sequences can be thought of as functions whose domain is the set of integers. The book contains eight practice tests five practice tests for Calculus AB and three practice tests for Calculus BC. 12 0 obj 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 All other trademarks and copyrights are the property of their respective owners. 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 . >> 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. Choose your answer to the question and click 'Continue' to see how you did. Ex 11.1.3 Determine whether \(\{\sqrt{n+47}-\sqrt{n}\}_{n=0}^{\infty}\) converges or diverges. Proofs for both tests are also given. /FontDescriptor 8 0 R stream Ex 11.7.2 Compute \(\lim_{n\to\infty} |a_{n+1}/a_n|\) for the series \(\sum 1/n\). 1000 1000 777.8 777.8 1000 1000 500 500 1000 1000 1000 777.8 1000 1000 611.1 611.1 Don't all infinite series grow to infinity? Solution. 4 avwo/MpLv) _C>5p*)i=^m7eE. 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 11.2.6 Compute \(\sum_{n=0}^\infty {4^{n+1}\over 5^n}\). S.QBt'(d|/"XH4!qbnEriHX)Gs2qN/G jq8$$< 1 2, 1 4, 1 8, Sequences of values of this type is the topic of this rst section. (answer), Ex 11.2.2 Explain why \(\sum_{n=1}^\infty {5\over 2^{1/n}+14}\) diverges. /Filter[/FlateDecode] (answer). Ex 11.7.1 Compute \(\lim_{n\to\infty} |a_{n+1}/a_n|\) for the series \(\sum 1/n^2\). Level up on all the skills in this unit and collect up to 2000 Mastery points! >> << Which of the following sequences is NOT a geometric sequence? 979.2 489.6 489.6 489.6] 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. 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 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 Which of the following is the 14th term of the sequence below? 777.8 777.8] 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 . 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 What is the sum of all the even integers from 2 to 250? 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). stream /FirstChar 0 << >> Harmonic series and p-series. /Length 200 590.3 885.4 885.4 295.1 324.7 531.3 531.3 531.3 531.3 531.3 795.8 472.2 531.3 767.4 We will also give many of the basic facts and properties well need as we work with sequences. When you have completed the free practice test, click 'View Results' to see your results. /Type/Font Learn for free about math, art, computer programming, economics, physics, chemistry, biology, medicine, finance, history, and more. 666.7 1000 1000 1000 1000 1055.6 1055.6 1055.6 777.8 666.7 666.7 450 450 450 450 Use the Comparison Test to determine whether each series in exercises 1 - 13 converges or diverges. Choose your answer to the question and click 'Continue' to see how you did. /Widths[611.8 816 761.6 679.6 652.8 734 707.2 761.6 707.2 761.6 707.2 571.2 544 544 441.3 461.2 353.6 557.3 473.4 699.9 556.4 477.4 454.9 312.5 377.9 623.4 489.6 272] Worksheets. %PDF-1.5 (answer). /FirstChar 0 Which is the finite sequence of four multiples of 9, starting with 9? Choose your answer to the question and click 'Continue' to see how you did. 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 >> (answer). Learn how this is possible, how we can tell whether a series converges, and how we can explore convergence in Taylor and Maclaurin series. /Type/Font My calculus 2 exam on sequence, infinite series & power seriesThe exam: https://bit.ly/36OHYcsAll the convergence tests: https://bit.ly/2IzqokhBest friend an. 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 see how we can use the first few terms of a power series to approximate a function. 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 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. Some infinite series converge to a finite value. Our mission is to provide a free, world-class education to anyone, anywhere. Then click 'Next Question' to answer the next question. Our mission is to provide a free, world-class education to anyone, anywhere. Which of the following sequences is NOT a geometric sequence? 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. x[[o6~cX/e`ElRm'1%J$%v)tb]1U2sRV}.l%s\Y UD+q}O+J Sequences & Series in Calculus Chapter Exam. /Filter /FlateDecode Accessibility StatementFor more information contact us [email protected]. /BaseFont/SFGTRF+CMSL12 The Alternating Series Test can be used only if the terms of the series alternate in sign. Published by Wiley. Comparison tests. 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 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? How many bricks are in the 12th row? A proof of the Alternating Series Test is also given. in calculus coursesincluding Calculus, Calculus II, Calculus III, AP Calculus and Precalculus. /Name/F2 endobj 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. If it converges, compute the limit. Each term is the sum of the previous two terms. Infinite series are sums of an infinite number of terms. (answer), Ex 11.1.6 Determine whether \(\left\{{2^n\over n! /FontDescriptor 20 0 R Then we can say that the series diverges without having to do any extra work. Divergence Test. The Integral Test can be used on a infinite series provided the terms of the series are positive and decreasing. 1) \(\displaystyle \sum^_{n=1}a_n\) where \(a_n=\dfrac{2}{n . Given that n=0 1 n3 +1 = 1.6865 n = 0 1 n 3 + 1 = 1.6865 determine the value of n=2 1 n3 +1 . If you're seeing this message, it means we're having trouble loading external resources on our website. Parametric equations, polar coordinates, and vector-valued functions Calculator-active practice: Parametric equations, polar coordinates, . Question 5 5. Choose your answer to the question and click 'Continue' to see how you did. However, use of this formula does quickly illustrate how functions can be represented as a power series.
