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. /FirstChar 0 /Type/Font The steps are terms in the sequence. At this time, I do not offer pdf's for solutions to individual problems. (answer), Ex 11.2.8 Compute \(\sum_{n=1}^\infty \left({3\over 5}\right)^n\). 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] )^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 \(\ln 2\). /LastChar 127 507.9 433.7 395.4 427.7 483.1 456.3 346.1 563.7 571.2 589.1 483.8 427.7 555.4 505 1000 1000 777.8 777.8 1000 1000 500 500 1000 1000 1000 777.8 1000 1000 611.1 611.1 The practice tests are composed (answer), Ex 11.2.9 Compute \(\sum_{n=1}^\infty {3^n\over 5^{n+1}}\). 611.8 897.2 734 761.6 666.2 761.6 720.6 544 707.2 734 734 1006 734 734 598.4 272 How many bricks are in the 12th row? /FirstChar 0 Then click 'Next Question' to answer the next question. Alternating series test. The numbers used come from a sequence. 21 0 obj 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 . Our mission is to provide a free, world-class education to anyone, anywhere. If it converges, compute the limit. AP is a registered trademark of the College Board, which has not reviewed this resource. 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. Which is the finite sequence of four multiples of 9, starting with 9? in calculus coursesincluding Calculus, Calculus II, Calculus III, AP Calculus and Precalculus. Series are sums of multiple terms. Strip out the first 3 terms from the series n=1 2n n2 +1 n = 1 2 n n 2 + 1. << (answer). . 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. Consider the series n a n. Divergence Test: If lim n a n 0, then n a n diverges. At this time, I do not offer pdfs for solutions to individual problems. Remark. Proofs for both tests are also given. (answer), Ex 11.9.2 Find a power series representation for \(1/(1-x)^2\). To log in and use all the features of Khan Academy, please enable JavaScript in your browser. (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. Premium members get access to this practice exam along with our entire library of lessons taught by subject matter experts. Premium members get access to this practice exam along with our entire library of lessons taught by subject matter experts. Here are a set of practice problems for the Series and Sequences chapter of the Calculus II notes. Determine whether the series is convergent or divergent. Published by Wiley. 15 0 obj Don't all infinite series grow to infinity? endobj 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 /FirstChar 0 555.6 577.8 577.8 597.2 597.2 736.1 736.1 527.8 527.8 583.3 583.3 583.3 583.3 750 Which of the following is the 14th term of the sequence below? 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. /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 Alternating Series Test In this section we will discuss using the Alternating Series Test to determine if an infinite series converges or diverges. 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. 1 2 + 1 4 + 1 8 + = n=1 1 2n = 1 We will need to be careful, but it turns out that we can . 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. 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. Sequences and Numerical series. stream Here are a set of practice problems for the Series and Sequences chapter of the Calculus II notes. The Alternating Series Test can be used only if the terms of the (b) Section 10.3 : Series - Basics. Bottom line -- series are just a lot of numbers added together. /BaseFont/PSJLQR+CMEX10 << 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 In exercises 3 and 4, do not attempt to determine whether the endpoints are in the interval of convergence. Ex 11.8.1 \(\sum_{n=0}^\infty n x^n\) (answer), Ex 11.8.2 \(\sum_{n=0}^\infty {x^n\over n! 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 . n = 1 n2 + 2n n3 + 3n2 + 1. /Filter /FlateDecode 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. Ratio test. Then click 'Next Question' to answer the next question. All other trademarks and copyrights are the property of their respective owners. \ _* %l~G"tytO(J*l+X@ uE: m/ ~&Q24Nss(7F!ky=4 Mijo8t;v If you're seeing this message, it means we're having trouble loading external resources on our website. << In order to use either test the terms of the infinite series must be positive. Which one of these sequences is a finite sequence? Psychological Research & Experimental Design, All Teacher Certification Test Prep Courses. Derivatives, Integrals, Sequences & Series, and Vector Valued Functions. S.QBt'(d|/"XH4!qbnEriHX)Gs2qN/G jq8$$< Here are a set of practice problems for the Series and Sequences chapter of the Calculus II notes. Strip out the first 3 terms from the series \( \displaystyle \sum\limits_{n = 1}^\infty {\frac{{{2^{ - n}}}}{{{n^2} + 1}}} \). Determine whether the following series converge or diverge. We will determine if a sequence in an increasing sequence or a decreasing sequence and hence if it is a monotonic sequence. %PDF-1.2 n a n converges if and only if the integral 1 f ( x) d x converges. endobj 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. >> If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. << 611.8 897.2 734 761.6 666.2 761.6 720.6 544 707.2 734 734 1006 734 734 598.4 272 /BaseFont/VMQJJE+CMR8 (answer). Choose your answer to the question and click 'Continue' to see how you did. (answer). endobj << Learn how this is possible, how we can tell whether a series converges, and how we can explore convergence in . Therefore the radius of convergence is R = , and the interval of convergence is ( - , ). xTn0+,ITi](N@ fH2}W"UG'.% Z#>y{!9kJ+ %PDF-1.5 (answer), Ex 11.9.3 Find a power series representation for \( 2/(1-x)^3\). Determine whether the series converge or diverge. After each bounce, the ball reaches a height that is 2/3 of the height from which it previously fell. What is the 83rd term of the sequence 91, 87, 83, 79, ( = a. UcTIjeB#vog-TM'FaTzG(:k-BNQmbj}'?^h<=XgS/]o4Ilv%Jm endstream We also discuss differentiation and integration of power series. Here are a set of practice problems for the Series and Sequences chapter of the Calculus II notes. << When you have completed the free practice test, click 'View Results' to see your results. 979.2 489.6 489.6 489.6] About this unit. 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. Which of the following represents the distance the ball bounces from the first to the seventh bounce with sigma notation? Calculus II-Sequences and Series. Ex 11.7.2 Compute \(\lim_{n\to\infty} |a_{n+1}/a_n|\) for the series \(\sum 1/n\). 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 /LastChar 127 These are homework exercises to accompany David Guichard's "General Calculus" Textmap. 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. All rights reserved. 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. /FontDescriptor 17 0 R 777.8 444.4 444.4 444.4 611.1 777.8 777.8 777.8 777.8] /Type/Font Sequences & Series in Calculus Chapter Exam - Study.com Integral Test: If a n = f ( n), where f ( x) is a non-negative non-increasing function, then. In the previous section, we determined the convergence or divergence of several series by . Series | Calculus 2 | Math | Khan Academy When you have completed the free practice test, click 'View Results' to see your results. Divergence Test. Absolute and conditional convergence. 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 Then click 'Next Question' to answer the . Each term is the difference of the previous two terms. endstream You may also use any of these materials for practice. Level up on all the skills in this unit and collect up to 2000 Mastery points! What is the radius of convergence? 777.8 777.8] 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. &/ r (answer), Ex 11.4.6 Approximate \(\sum_{n=1}^\infty (-1)^{n-1}{1\over n^4}\) to two decimal places. Math Journey: Calculus, ODEs, Linear Algebra and Beyond nth-term test. >> 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. Ex 11.1.3 Determine whether {n + 47 n} . 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. A brick wall has 60 bricks in the first row, but each row has 3 fewer bricks than the previous one. /BaseFont/UNJAYZ+CMR12 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 Level up on all the skills in this unit and collect up to 2000 Mastery points! /Name/F3 endobj When you have completed the free practice test, click 'View Results' to see your results. Ex 11.7.1 Compute \(\lim_{n\to\infty} |a_{n+1}/a_n|\) for the series \(\sum 1/n^2\). 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. %PDF-1.5 % }\) (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! 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. /Subtype/Type1 Calculus II - Series & Sequences (Practice Problems) - Lamar University PDF Arithmetic Sequences And Series Practice Problems Choose the equation below that represents the rule for the nth term of the following geometric sequence: 128, 64, 32, 16, 8, . Worksheets. (a) $\sum_{n=1}^{\infty} \frac{(-1)^n}{\sqrt{n}}$ (b) $\sum_{n=1}^{\infty}(-1)^n \frac{n}{2 n-1}$ SAT Practice Questions- All Maths; SAT Practice Test Questions- Reading , Writing and Language; KS 1-2 Math, Science and SAT . stream 8 0 obj Choosing a Convergence Test | Calculus II - Lumen Learning Harmonic series and p-series. 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 1. 21 terms. 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. /Name/F1 Other sets by this creator. 18 0 obj To use integration by parts in Calculus, follow these steps: Decompose the entire integral (including dx) into two factors. (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. /FontDescriptor 11 0 R copyright 2003-2023 Study.com. copyright 2003-2023 Study.com. 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 What if the interval is instead \([1,3/2]\)? endobj Use the Comparison Test to determine whether each series in exercises 1 - 13 converges or diverges. /LastChar 127 Comparison Test: This applies . ,vEmO8/OuNVRaLPqB.*l. /Length 200 /FirstChar 0 5.3.1 Use the divergence test to determine whether a series converges or diverges. xWKoFWlojCpP NDED$(lq"g|3g6X_&F1BXIM5d gOwaN9c,r|9 5.3 The Divergence and Integral Tests - Calculus Volume 2 - OpenStax %PDF-1.5 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? Then click 'Next Question' to answer the next question. /Filter /FlateDecode We will examine Geometric Series, Telescoping Series, and Harmonic Series. 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 When given a sum a[n], if the limit as n-->infinity does not exist or does not equal 0, the sum diverges. Ex 11.9.5 Find a power series representation for \(\int\ln(1-x)\,dx\). Integral test. Ex 11.7.4 Compute \(\lim_{n\to\infty} |a_n|^{1/n}\) for the series \(\sum 1/n\). 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 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. 31 terms. We will also give many of the basic facts and properties well need as we work with sequences. Ex 11.1.1 Compute \(\lim_{x\to\infty} x^{1/x}\). Ex 11.10.8 Find the first four terms of the Maclaurin series for \(\tan x\) (up to and including the \( x^3\) term). %|S#?\A@D-oS)lW=??nn}y]Tb!!o_=;]ha,J[. 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 << 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@ " Which of the following sequences is NOT a geometric sequence? Mediansandsuch - Medians - MATH 126 Medians and Such Let X be - Studocu Question 5 5. 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, \( \displaystyle \sum\limits_{n = 1}^\infty {\left( {n{2^n} - {3^{1 - n}}} \right)} \), \( \displaystyle \sum\limits_{n = 7}^\infty {\frac{{4 - n}}{{{n^2} + 1}}} \), \( \displaystyle \sum\limits_{n = 2}^\infty {\frac{{{{\left( { - 1} \right)}^{n - 3}}\left( {n + 2} \right)}}{{{5^{1 + 2n}}}}} \). >> Which equation below represents a geometric sequence? )^2\over n^n}\) (answer). Root Test In this section we will discuss using the Root Test to determine if an infinite series converges absolutely or diverges. Learning Objectives. 9 0 obj << 26 0 obj Contact us by phone at (877)266-4919, or by mail at 100ViewStreet#202, MountainView, CA94041. If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. /Filter /FlateDecode (answer), Ex 11.9.4 Find a power series representation for \( 1/(1-x)^3\). /Name/F5 4 avwo/MpLv) _C>5p*)i=^m7eE. (answer), Ex 11.2.7 Compute \(\sum_{n=0}^\infty {3^{n+1}\over 7^{n+1}}\). 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. Some infinite series converge to a finite value. Martha_Austin Teacher. (1 point) Is the integral Z 1 1 1 x2 dx an improper integral? For each of the following series, determine which convergence test is the best to use and explain why. Convergence/Divergence of Series In this section we will discuss in greater detail the convergence and divergence of infinite series. We will also see how we can use the first few terms of a power series to approximate a function. Sequences and Series for Calculus Chapter Exam - Study.com Your instructor might use some of these in class. Images. 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_. Note that some sections will have more problems than others and some will have more or less of a variety of problems. We will also determine a sequence is bounded below, bounded above and/or bounded. 68 0 obj (answer), Ex 11.10.10 Use a combination of Maclaurin series and algebraic manipulation to find a series centered at zero for \( xe^{-x}\). %%EOF 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. /Type/Font Determine whether each series converges absolutely, converges conditionally, or diverges. The chapter headings refer to Calculus, Sixth Edition by Hughes-Hallett et al. Research Methods Midterm. Taylor Series In this section we will discuss how to find the Taylor/Maclaurin Series for a function. The following is a list of worksheets and other materials related to Math 129 at the UA. 531.3 531.3 531.3] Math 129 - Calculus II. 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. /Subtype/Type1 /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 >> Maclaurin series of e, sin(x), and cos(x). /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 Chapter 10 : Series and Sequences. Which of the following sequences follows this formula. (answer), Ex 11.2.4 Compute \(\sum_{n=0}^\infty {4\over (-3)^n}- {3\over 3^n}\). (answer), Ex 11.2.3 Explain why \(\sum_{n=1}^\infty {3\over n}\) diverges. To log in and use all the features of Khan Academy, please enable JavaScript in your browser. /FirstChar 0 Then click 'Next Question' to answer the next question. We also derive some well known formulas for Taylor series of \({\bf e}^{x}\) , \(\cos(x)\) and \(\sin(x)\) around \(x=0\). >> Khan Academy is a 501(c)(3) nonprofit organization. 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 For each function, find the Maclaurin series or Taylor series centered at $a$, and the radius of convergence. Math 106 (Calculus II): old exams. Infinite series are sums of an infinite number of terms. Complementary General calculus exercises can be found for other Textmaps and can be accessed here. 2.(a). Sequences & Series in Calculus Chapter Exam. If you . Calculus 2 | Math | Khan Academy endobj !A1axw)}p]WgxmkFftu 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. 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 . Integral Test In this section we will discuss using the Integral Test to determine if an infinite series converges or diverges. /Filter /FlateDecode Which is the infinite sequence starting with 1 where each number is the previous number times 3? Which of the following sequences follows this formula? stream /Length 465 >> /Name/F6 Math 1242: Calculus II - University of North Carolina at Charlotte 24 0 obj Good luck! Math 106 (Calculus II): old exams | Mathematics | Bates College If it con-verges, nd the limit. /FontDescriptor 23 0 R (answer), Ex 11.2.5 Compute \(\sum_{n=0}^\infty {3\over 2^n}+ {4\over 5^n}\). 45 0 obj 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 PDF Calc II: Practice Final Exam - Columbia University If it converges, compute the limit. 207 0 obj <> endobj endstream endobj startxref stream 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. (answer), Ex 11.1.5 Determine whether \(\left\{{n+47\over\sqrt{n^2+3n}}\right\}_{n=1}^{\infty}\) converges or diverges. Part II. (answer), Ex 11.10.9 Use a combination of Maclaurin series and algebraic manipulation to find a series centered at zero for \( x\cos (x^2)\). We will also give many of the basic facts, properties and ways we can use to manipulate a series. We will focus on the basic terminology, limits of sequences and convergence of sequences in this section. /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 /LastChar 127 Strategies for Testing Series - University of Texas at Austin
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