PDF FINAL EXAM CALCULUS 2 - Department of Mathematics 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 stream (answer), Ex 11.2.8 Compute \(\sum_{n=1}^\infty \left({3\over 5}\right)^n\). Alternating Series Test - In this section we will discuss using the Alternating Series Test to determine if an infinite series converges or diverges. /Length 2492 How many bricks are in the 12th row? %PDF-1.5 (answer), Ex 11.9.4 Find a power series representation for \( 1/(1-x)^3\). 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 What if the interval is instead \([1,3/2]\)? 11.E: Sequences and Series (Exercises) - Mathematics LibreTexts /Subtype/Type1 Learning Objectives. To log in and use all the features of Khan Academy, please enable JavaScript in your browser. %|S#?\A@D-oS)lW=??nn}y]Tb!!o_=;]ha,J[. Calculus 2. Good luck! /Type/Font In exercises 3 and 4, do not attempt to determine whether the endpoints are in the interval of convergence. Series The Basics In this section we will formally define an infinite series. 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. Which of the following is the 14th term of the sequence below? Each term is the product of the two previous terms. xWKoFWlojCpP NDED$(lq"g|3g6X_&F1BXIM5d gOwaN9c,r|9 Choose your answer to the question and click 'Continue' to see how you did. Most sections should have a range of difficulty levels in the problems although this will vary from section to section. Harmonic series and p-series. Comparison tests. If it converges, compute the limit. Which of the following sequences is NOT a geometric sequence? endstream Math 106 (Calculus II): old exams | Mathematics | Bates College Parametric equations, polar coordinates, and vector-valued functions Calculator-active practice: Parametric equations, polar coordinates, . Question 5 5. Then click 'Next Question' to answer the next question. (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. Strategies for Testing Series - University of Texas at Austin /Name/F6 Remark. /Name/F2 Study Online AP Calculus AB and BC: Chapter 9 -Infinite Sequences and Series : 9.2 -The Integral Test and p-Series Study Notes Prepared by AP Teachers Skip to content . >> Ex 11.1.3 Determine whether \(\{\sqrt{n+47}-\sqrt{n}\}_{n=0}^{\infty}\) converges or diverges. Strip out the first 3 terms from the series n=1 2n n2 +1 n = 1 2 n n 2 + 1. Sequences and Numerical series. 62 0 obj ZrNRG{I~(iw%0W5b)8*^ yyCCy~Cg{C&BPsTxp%p >> Some infinite series converge to a finite value. If you're seeing this message, it means we're having trouble loading external resources on our website. 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 << Strip out the first 3 terms from the series \( \displaystyle \sum\limits_{n = 1}^\infty {\frac{{{2^{ - n}}}}{{{n^2} + 1}}} \). We also derive some well known formulas for Taylor series of \({\bf e}^{x}\) , \(\cos(x)\) and \(\sin(x)\) around \(x=0\). Quiz 1: 5 questions Practice what you've learned, and level up on the above skills. PDF Review Sheet for Calculus 2 Sequences and Series - Derrick Chung Part II. % n = 1 n2 + 2n n3 + 3n2 + 1. (a) $\sum_{n=1}^{\infty} \frac{(-1)^n}{\sqrt{n}}$ (b) $\sum_{n=1}^{\infty}(-1)^n \frac{n}{2 n-1}$ A ball is dropped from an unknown height (h) and it repeatedly bounces on the floor. (answer), Ex 11.2.6 Compute \(\sum_{n=0}^\infty {4^{n+1}\over 5^n}\). 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. << Don't all infinite series grow to infinity? x[[o6~cX/e`ElRm'1%J$%v)tb]1U2sRV}.l%s\Y UD+q}O+J The following is a list of worksheets and other materials related to Math 129 at the UA. The book contains eight practice tests five practice tests for Calculus AB and three practice tests for Calculus BC. Ex 11.7.2 Compute \(\lim_{n\to\infty} |a_{n+1}/a_n|\) for the series \(\sum 1/n\). 5.3.3 Estimate the value of a series by finding bounds on its remainder term. SAT Practice Questions- All Maths; SAT Practice Test Questions- Reading , Writing and Language; KS 1-2 Math, Science and SAT . Donate or volunteer today! << PDF Calculus II Series - Things to Consider - California State University (answer), Ex 11.2.2 Explain why \(\sum_{n=1}^\infty {5\over 2^{1/n}+14}\) diverges. PDF M 172 - Calculus II - Chapter 10 Sequences and Series %PDF-1.5 % Legal. /Name/F4 Calculus II For Dummies Cheat Sheet - dummies 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 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). &/ r |: The Ratio Test shows us that regardless of the choice of x, the series converges. xTn0+,ITi](N@ fH2}W"UG'.% Z#>y{!9kJ+ 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 11.9.5 Find a power series representation for \(\int\ln(1-x)\,dx\). The Alternating Series Test can be used only if the terms of the 26 0 obj When given a sum a[n], if the limit as n-->infinity does not exist or does not equal 0, the sum diverges. (answer). We will also see how we can use the first few terms of a power series to approximate a function. Here are a set of practice problems for the Series and Sequences chapter of the Calculus II notes. >> Integral Test In this section we will discuss using the Integral Test to determine if an infinite series converges or diverges. Use the Comparison Test to determine whether each series in exercises 1 - 13 converges or diverges. stream 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. 11.E: Sequences and Series (Exercises) These are homework exercises to accompany David Guichard's "General Calculus" Textmap. 252 0 obj <>stream 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 )^2\over n^n}\) (answer). Published by Wiley. >> %PDF-1.2 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, \( \displaystyle \sum\limits_{n = 1}^\infty {\left( {n{2^n} - {3^{1 - n}}} \right)} \), \( \displaystyle \sum\limits_{n = 7}^\infty {\frac{{4 - n}}{{{n^2} + 1}}} \), \( \displaystyle \sum\limits_{n = 2}^\infty {\frac{{{{\left( { - 1} \right)}^{n - 3}}\left( {n + 2} \right)}}{{{5^{1 + 2n}}}}} \). >> 1. endstream xu? ~k"xPeEV4Vcwww \ a:5d*%30EU9>,e92UU3Voj/$f BS!.eSloaY&h&Urm!U3L%g@'>`|$ogJ We will also give the Divergence Test for series in this section. If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. Good luck! % OR sequences are lists of numbers, where the numbers may or may not be determined by a pattern. Free Practice Test Instructions: Choose your answer to the question and click 'Continue' to see how you did. Then click 'Next Question' to answer the next question. 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 . To use the Geometric Series formula, the function must be able to be put into a specific form, which is often impossible. Integral Test: If a n = f ( n), where f ( x) is a non-negative non-increasing function, then. (answer). 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. All rights reserved. The chapter headings refer to Calculus, Sixth Edition by Hughes-Hallett et al. 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 . /BaseFont/BPHBTR+CMMI12 Solving My Calc 2 Exam#3 (Sequence, Infinite Series & Power Series) (answer), Ex 11.2.9 Compute \(\sum_{n=1}^\infty {3^n\over 5^{n+1}}\). 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! /BaseFont/CQGOFL+CMSY10 These are homework exercises to accompany David Guichard's "General Calculus" Textmap. Choosing a Convergence Test | Calculus II - Lumen Learning 6.5E: Exercises for Comparison Test - Mathematics LibreTexts A Lot of Series Test Practice Problems - YouTube To use integration by parts in Calculus, follow these steps: Decompose the entire integral (including dx) into two factors. Math Journey: Calculus, ODEs, Linear Algebra and Beyond We will examine Geometric Series, Telescoping Series, and Harmonic Series. 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. Bottom line -- series are just a lot of numbers added together. If you're seeing this message, it means we're having trouble loading external resources on our website. Which of the following sequences is NOT a geometric sequence? Convergence/Divergence of Series In this section we will discuss in greater detail the convergence and divergence of infinite series. }\right\}_{n=0}^{\infty}\) converges or diverges. Series Infinite geometric series: Series nth-term test: Series Integral test: Series Harmonic series and p-series: Series Comparison tests: . (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. Therefore the radius of convergence is R = , and the interval of convergence is ( - , ). 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. endobj 68 0 obj 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. MATH 126 Medians and Such. Accessibility StatementFor more information contact us atinfo@libretexts.org. endobj 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 In order to use either test the terms of the infinite series must be positive. 5.3.2 Use the integral test to determine the convergence of a series. n a n converges if and only if the integral 1 f ( x) d x converges. 17 0 obj >> When you have completed the free practice test, click 'View Results' to see your results. Sequences & Series in Calculus Chapter Exam. 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. (b) 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. Determine whether each series converges absolutely, converges conditionally, or diverges. What is the sum of all the even integers from 2 to 250? A proof of the Ratio Test is also given. copyright 2003-2023 Study.com. Proofs for both tests are also given. \ _* %l~G"tytO(J*l+X@ uE: m/ ~&Q24Nss(7F!ky=4 Mijo8t;v /LastChar 127 << /Length 569 Given that \( \displaystyle \sum\limits_{n = 0}^\infty {\frac{1}{{{n^3} + 1}}} = 1.6865\) determine the value of \( \displaystyle \sum\limits_{n = 2}^\infty {\frac{1}{{{n^3} + 1}}} \). What is the radius of convergence? When you have completed the free practice test, click 'View Results' to see your results. /Subtype/Type1 endobj /Type/Font . 24 0 obj L7s[AQmT*Z;HK%H0yqt1r8 PDF Read Free Answers To Algebra 2 Practice B Ellipses