nn = 0. Free Practice Test Instructions: Choose your answer to the question and click 'Continue' to see how you did. Chapter 10 : Series and Sequences. >> 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), Ex 11.9.4 Find a power series representation for \( 1/(1-x)^3\). >> 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. Ex 11.10.8 Find the first four terms of the Maclaurin series for \(\tan x\) (up to and including the \( x^3\) term). Proofs for both tests are also given. . /LastChar 127 We will also give many of the basic facts and properties well need as we work with sequences. If it converges, compute the limit. 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 1277.8 555.6 1000 1444.4 555.6 1000 1444.4 472.2 472.2 527.8 527.8 527.8 527.8 666.7 722.6 693.1 833.5 795.8 382.6 545.5 825.4 663.6 972.9 795.8 826.4 722.6 826.4 781.6 Estimating the Value of a Series In this section we will discuss how the Integral Test, Comparison Test, Alternating Series Test and the Ratio Test can, on occasion, be used to estimating the value of an infinite series. Each term is the product of the two previous terms. We also discuss differentiation and integration of power series. We also derive some well known formulas for Taylor series of \({\bf e}^{x}\) , \(\cos(x)\) and \(\sin(x)\) around \(x=0\). 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 To log in and use all the features of Khan Academy, please enable JavaScript in your browser. /Length 465 Ex 11.1.2 Use the squeeze theorem to show that limn n! % /Type/Font >> /Type/Font We will also give the Divergence Test for series in this section. Which one of these sequences is a finite sequence? stream )^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 \(\ln 2\). 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. Quiz 1: 5 questions Practice what you've learned, and level up on the above skills. All rights reserved. These are homework exercises to accompany David Guichard's "General Calculus" Textmap. 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. In exercises 3 and 4, do not attempt to determine whether the endpoints are in the interval of convergence. AP is a registered trademark of the College Board, which has not reviewed this resource. How many bricks are in the 12th row? Taylor Series In this section we will discuss how to find the Taylor/Maclaurin Series for a function. Maclaurin series of e, sin(x), and cos(x). Our mission is to provide a free, world-class education to anyone, anywhere. Level up on all the skills in this unit and collect up to 2000 Mastery points! What is the radius of convergence? We will also see how we can use the first few terms of a power series to approximate a function. /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 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. Alternating Series Test - In this section we will discuss using the Alternating Series Test to determine if an infinite series converges or diverges. (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 \(\sum_{n=0}^N {1\over e^n} + 10^{-4}\). 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. 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. Strip out the first 3 terms from the series n=1 2n n2 +1 n = 1 2 n n 2 + 1. Integral Test: If a n = f ( n), where f ( x) is a non-negative non-increasing function, then. /FirstChar 0 Here are a set of practice problems for the Series and Sequences chapter of the Calculus II notes. Solution. /Name/F6 /Filter /FlateDecode (answer). 45 0 obj >> 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. 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 )^2\over n^n}\) (answer). << (answer), Ex 11.10.10 Use a combination of Maclaurin series and algebraic manipulation to find a series centered at zero for \( xe^{-x}\). Learn how this is possible, how we can tell whether a series converges, and how we can explore convergence in Taylor and Maclaurin series. 611.8 897.2 734 761.6 666.2 761.6 720.6 544 707.2 734 734 1006 734 734 598.4 272 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. %%EOF /FirstChar 0 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. Applications of Series In this section we will take a quick look at a couple of applications of series. %PDF-1.5 /FontDescriptor 14 0 R . Math 129 - Calculus II. /Subtype/Type1 666.7 1000 1000 1000 1000 1055.6 1055.6 1055.6 777.8 666.7 666.7 450 450 450 450 /Length 1247 >> Series Infinite geometric series: Series nth-term test: Series Integral test: Series Harmonic series and p-series: Series Comparison tests: . All rights reserved. Learn how this is possible, how we can tell whether a series converges, and how we can explore convergence in . %PDF-1.5 % /Length 200 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 . 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 0 xYKs6W(MCG:9iIO=(lkFRI$x$AMN/" J?~i~d cXf9o/r.&Lxy%/D-Yt+"LX]Sfp]Xl-aM_[6(*~mQbh*28AjZx0 =||. 18 0 obj (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. Series The Basics In this section we will formally define an infinite series. (answer), Ex 11.9.2 Find a power series representation for \(1/(1-x)^2\). Determine whether each series converges absolutely, converges conditionally, or diverges. /Subtype/Type1 SAT Practice Questions- All Maths; SAT Practice Test Questions- Reading , Writing and Language; KS 1-2 Math, Science and SAT . /FontDescriptor 11 0 R /FontDescriptor 8 0 R 21 0 obj /BaseFont/CQGOFL+CMSY10 The Alternating Series Test can be used only if the terms of the series alternate in sign. %|S#?\A@D-oS)lW=??nn}y]Tb!!o_=;]ha,J[. 668.3 724.7 666.7 666.7 666.7 666.7 666.7 611.1 611.1 444.4 444.4 444.4 444.4 500 805.6 805.6 1277.8 1277.8 811.1 811.1 875 875 666.7 666.7 666.7 666.7 666.7 666.7 /FontDescriptor 23 0 R UcTIjeB#vog-TM'FaTzG(:k-BNQmbj}'?^h<=XgS/]o4Ilv%Jm }\right\}_{n=0}^{\infty}\) converges or diverges. << In order to use either test the terms of the infinite series must be positive. 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. Infinite series are sums of an infinite number of terms. If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. 207 0 obj <> endobj The Alternating Series Test can be used only if the terms of the Complementary General calculus exercises can be found for other Textmaps and can be accessed here. Don't all infinite series grow to infinity? Example 1. 2.(a). 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 \(\sum_{n=1}^N {1\over n^4} + 0.005\). A ball is dropped from an unknown height (h) and it repeatedly bounces on the floor. (a) $\sum_{n=1}^{\infty} \frac{(-1)^n}{\sqrt{n}}$ (b) $\sum_{n=1}^{\infty}(-1)^n \frac{n}{2 n-1}$ Most sections should have a range of difficulty levels in the problems although this will vary from section to section. Series are sums of multiple terms. n = 1 n 2 + 2 n n 3 + 3 n . ,vEmO8/OuNVRaLPqB.*l. endstream endobj 208 0 obj <. Determine whether the sequence converges or diverges. Ex 11.7.3 Compute \(\lim_{n\to\infty} |a_n|^{1/n}\) for the series \(\sum 1/n^2\). nth-term test. 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. /Length 569 Learn for free about math, art, computer programming, economics, physics, chemistry, biology, medicine, finance, history, and more. << Which of the following sequences is NOT a geometric sequence? endobj /Filter /FlateDecode 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 /BaseFont/PSJLQR+CMEX10 It turns out the answer is no. /Type/Font endstream endobj startxref 15 0 obj 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. xu? ~k"xPeEV4Vcwww \ a:5d*%30EU9>,e92UU3Voj/$f BS!.eSloaY&h&Urm!U3L%g@'>`|$ogJ 5.3.1 Use the divergence test to determine whether a series converges or diverges. Harmonic series and p-series. /Type/Font Choose your answer to the question and click 'Continue' to see how you did. /Length 2492 Then click 'Next Question' to answer the next question. We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. 70 terms. Complementary General calculus exercises can be found for other Textmaps and can be accessed here. 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 xWKoFWlojCpP NDED$(lq"g|3g6X_&F1BXIM5d gOwaN9c,r|9 << 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 Here are a set of practice problems for the Series and Sequences chapter of the Calculus II notes. Calculus 2. 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. /LastChar 127 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. What is the 83rd term of the sequence 91, 87, 83, 79, ( = a. Khan Academy is a 501(c)(3) nonprofit organization. << If you're seeing this message, it means we're having trouble loading external resources on our website. After each bounce, the ball reaches a height that is 2/3 of the height from which it previously fell. n = 1 n2 + 2n n3 + 3n2 + 1. Final: all from 02/05 and 03/11 exams (except work, separation of variables, and probability) plus sequences, series, convergence tests, power series, Taylor series. endobj Strip out the first 3 terms from the series \( \displaystyle \sum\limits_{n = 1}^\infty {\frac{{{2^{ - n}}}}{{{n^2} + 1}}} \). Contact us by phone at (877)266-4919, or by mail at 100ViewStreet#202, MountainView, CA94041. Which of the following sequences follows this formula? Which of the following sequences is NOT a geometric sequence? Images. (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 \(\sum_{n=1}^N {\ln n\over n^2} + 0.005\). (answer). 1. If a geometric series begins with the following term, what would the next term be? >> If it converges, compute the limit. Donate or volunteer today! Ratio test. Determine whether the series converge or diverge. >> 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. When you have completed the free practice test, click 'View Results' to see your results. 1) \(\displaystyle \sum^_{n=1}a_n\) where \(a_n=\dfrac{2}{n . copyright 2003-2023 Study.com. Research Methods Midterm. Determine whether each series converges or 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_. If it con-verges, nd the limit. Psychological Research & Experimental Design, All Teacher Certification Test Prep Courses. 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 . We will also determine a sequence is bounded below, bounded above and/or bounded. Accessibility StatementFor more information contact us atinfo@libretexts.org. Infinite series are sums of an infinite number of terms. /Name/F5 Find the radius and interval of convergence for each of the following series: Solution (a) We apply the Ratio Test to the series n = 0 | x n n! Parametric equations, polar coordinates, and vector-valued functions Calculator-active practice: Parametric equations, polar coordinates, . 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. endobj Some infinite series converge to a finite value. /Filter /FlateDecode (5 points) Evaluate the integral: Z 1 1 1 x2 dx = SOLUTION: The function 1/x2 is undened at x = 0, so we we must evaluate the im- proper integral as a limit. 252 0 obj <>stream /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 Ex 11.9.5 Find a power series representation for \(\int\ln(1-x)\,dx\). 31 terms. << Ex 11.1.1 Compute \(\lim_{x\to\infty} x^{1/x}\). \ _* %l~G"tytO(J*l+X@ uE: m/ ~&Q24Nss(7F!ky=4 Mijo8t;v /Name/F3 Then click 'Next Question' to answer the . << 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 At this time, I do not offer pdf's for solutions to individual problems. We will focus on the basic terminology, limits of sequences and convergence of sequences in this section. << Calc II: Practice Final Exam 5 and our series converges because P nbn is a p-series with p= 4=3 >1: (b) X1 n=1 lnn n3 Set f(x) = lnx x3 and check that f0= 43x lnx+ x 4 <0 Consider the series n a n. Divergence Test: If lim n a n 0, then n a n diverges. Premium members get access to this practice exam along with our entire library of lessons taught by subject matter experts. We will also give many of the basic facts, properties and ways we can use to manipulate a series. endobj << /Subtype/Type1 &/ r The numbers used come from a sequence. 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. >> 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. MULTIPLE CHOICE: Circle the best answer. 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 illustrate how we can find a series representation for indefinite integrals that cannot be evaluated by any other method. /Widths[611.8 816 761.6 679.6 652.8 734 707.2 761.6 707.2 761.6 707.2 571.2 544 544 272 761.6 462.4 462.4 761.6 734 693.4 707.2 747.8 666.2 639 768.3 734 353.2 503 761.2 (answer). The chapter headings refer to Calculus, Sixth Edition by Hughes-Hallett et al. All other trademarks and copyrights are the property of their respective owners. Choose the equation below that represents the rule for the nth term of the following geometric sequence: 128, 64, 32, 16, 8, . Learning Objectives. The Integral Test can be used on a infinite series provided the terms of the series are positive and decreasing. Indiana Core Assessments Mathematics: Test Prep & Study Guide. copyright 2003-2023 Study.com. To use the Geometric Series formula, the function must be able to be put into a specific form, which is often impossible. Sequences & Series in Calculus Chapter Exam. 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 If L = 1, then the test is inconclusive. 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 The following is a list of worksheets and other materials related to Math 129 at the UA. Section 10.3 : Series - Basics. My calculus 2 exam on sequence, infinite series & power seriesThe exam: https://bit.ly/36OHYcsAll the convergence tests: https://bit.ly/2IzqokhBest friend an. Ex 11.7.2 Compute \(\lim_{n\to\infty} |a_{n+1}/a_n|\) for the series \(\sum 1/n\). /FirstChar 0 x=S0 Determine whether the following series converge or diverge. Which equation below represents a geometric sequence? 11.E: Sequences and Series (Exercises) These are homework exercises to accompany David Guichard's "General Calculus" Textmap. 777.8 777.8] About this unit. /Widths[777.8 277.8 777.8 500 777.8 500 777.8 777.8 777.8 777.8 777.8 777.8 777.8 531.3 590.3 560.8 414.1 419.1 413.2 590.3 560.8 767.4 560.8 560.8 472.2 531.3 1062.5 stream Good luck! (answer), Ex 11.2.3 Explain why \(\sum_{n=1}^\infty {3\over n}\) diverges. Ex 11.1.3 Determine whether {n + 47 n} . !A1axw)}p]WgxmkFftu Integral Test In this section we will discuss using the Integral Test to determine if an infinite series converges or diverges. Math 106 (Calculus II): old exams. Note that some sections will have more problems than others and some will have more or less of a variety of problems. Ex 11.7.4 Compute \(\lim_{n\to\infty} |a_n|^{1/n}\) for the series \(\sum 1/n\). 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 Choose your answer to the question and click 'Continue' to see how you did. S.QBt'(d|/"XH4!qbnEriHX)Gs2qN/G jq8$$< 4 avwo/MpLv) _C>5p*)i=^m7eE. /LastChar 127 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. }\) (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! Let the factor without dx equal u and the factor with dx equal dv. 531.3 531.3 531.3] 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. Sequences and Numerical series. /Type/Font If the series is an alternating series, determine whether it converges absolutely, converges conditionally, or diverges. Determine whether the series is convergent or divergent. |: The Ratio Test shows us that regardless of the choice of x, the series converges. A proof of the Root Test is also given. Some infinite series converge to a finite value. 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 . 24 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. turtle bay room service menu,