calculus 2 series and sequences practice test

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. All rights reserved. /BaseFont/PSJLQR+CMEX10 /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 Sequences & Series in Calculus Chapter Exam. 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. 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. Which of the following represents the distance the ball bounces from the first to the seventh bounce with sigma notation? >> A proof of the Ratio Test is also given. 531.3 531.3 531.3] (answer), Ex 11.2.5 Compute \(\sum_{n=0}^\infty {3\over 2^n}+ {4\over 5^n}\). Alternating series test. If it converges, compute the limit. 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 S.QBt'(d|/"XH4!qbnEriHX)Gs2qN/G jq8$$< Which of the sequences below has the recursive rule where each number is the previous number times 2? 590.3 767.4 795.8 795.8 1091 795.8 795.8 649.3 295.1 531.3 295.1 531.3 295.1 295.1 Choose your answer to the question and click 'Continue' to see how you did. (1 point) Is the integral Z 1 1 1 x2 dx an improper integral? 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. << Calculus II-Sequences and Series. The practice tests are composed 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 Given that \( \displaystyle \sum\limits_{n = 0}^\infty {\frac{1}{{{n^3} + 1}}} = 1.6865\) determine the value of \( \displaystyle \sum\limits_{n = 2}^\infty {\frac{1}{{{n^3} + 1}}} \). At this time, I do not offer pdf's for . /Subtype/Type1 stream /Subtype/Type1 /Length 1722 /Type/Font Math 106 (Calculus II): old exams. What is the radius of convergence? Choose your answer to the question and click 'Continue' to see how you did. (answer), Ex 11.2.6 Compute \(\sum_{n=0}^\infty {4^{n+1}\over 5^n}\). 207 0 obj <> endobj << 1 2, 1 4, 1 8, Sequences of values of this type is the topic of this rst section. Bottom line -- series are just a lot of numbers added together. 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. Each review chapter is packed with equations, formulas, and examples with solutions, so you can study smarter and score a 5! 31 terms. 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. 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. Integral Test: If a n = f ( n), where f ( x) is a non-negative non-increasing function, then. For problems 1 - 3 perform an index shift so that the series starts at n = 3 n = 3. 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. In order to use either test the terms of the infinite series must be positive. sCA%HGEH[ Ah)lzv<7'9&9X}xbgY[ xI9i,c_%tz5RUam\\6(ke9}Yv`B7yYdWrJ{KZVUYMwlbN_>[wle\seUy24P,PyX[+l\c $w^rvo]cYc@bAlfi6);;wOF&G_. >> >> /Length 2492 << Sequences and Series. endstream endobj startxref copyright 2003-2023 Study.com. Convergence/Divergence of Series In this section we will discuss in greater detail the convergence and divergence of infinite series. To use the Geometric Series formula, the function must be able to be put into a specific form, which is often impossible. /LastChar 127 Determine whether each series converges or diverges. << Ex 11.1.3 Determine whether {n + 47 n} . If you're seeing this message, it means we're having trouble loading external resources on our website. hb```9B 7N0$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@ " Quiz 2: 8 questions Practice what you've learned, and level up on the above skills. How many bricks are in the 12th row? Let the factor without dx equal u and the factor with dx equal dv. Infinite series are sums of an infinite number of terms. 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. Determine whether the series converge or diverge. stream /Filter /FlateDecode Level up on all the skills in this unit and collect up to 2000 Mastery points! 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). Legal. 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 Applications of Series In this section we will take a quick look at a couple of applications of series. 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, \( \displaystyle \sum\limits_{n = 1}^\infty {\left( {n{2^n} - {3^{1 - n}}} \right)} \), \( \displaystyle \sum\limits_{n = 7}^\infty {\frac{{4 - n}}{{{n^2} + 1}}} \), \( \displaystyle \sum\limits_{n = 2}^\infty {\frac{{{{\left( { - 1} \right)}^{n - 3}}\left( {n + 2} \right)}}{{{5^{1 + 2n}}}}} \). SAT Practice Questions- All Maths; SAT Practice Test Questions- Reading , Writing and Language; KS 1-2 Math, Science and SAT . )^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 \(\ln 2\). ZrNRG{I~(iw%0W5b)8*^ yyCCy~Cg{C&BPsTxp%p 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. Then we can say that the series diverges without having to do any extra work. !A1axw)}p]WgxmkFftu >> >> 12 0 obj Example 1. (answer). We will also give the Divergence Test for series in this section. Khan Academy is a nonprofit with the mission of providing a free, world-class education for anyone, anywhere. YesNo 2.(b). If it con-verges, nd the limit. Ex 11.7.9 Prove theorem 11.7.3, the root test. << Alternating Series Test In this section we will discuss using the Alternating Series Test to determine if an infinite series converges or diverges. %PDF-1.5 % endstream %PDF-1.5 If you . ,vEmO8/OuNVRaLPqB.*l. Ex 11.8.1 \(\sum_{n=0}^\infty n x^n\) (answer), Ex 11.8.2 \(\sum_{n=0}^\infty {x^n\over n! (answer), Ex 11.10.9 Use a combination of Maclaurin series and algebraic manipulation to find a series centered at zero for \( x\cos (x^2)\). 17 0 obj Maclaurin series of e, sin(x), and cos(x). /Filter /FlateDecode Consider the series n a n. Divergence Test: If lim n a n 0, then n a n diverges. x=S0 Good luck! endobj Each term is the product of the two previous terms. Calculus (single and multi-variable) Ordinary Differential equations (upto 2nd order linear with Laplace transforms, including Dirac Delta functions and Fourier Series. (answer). The steps are terms in the sequence. 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 More on Sequences In this section we will continue examining sequences. 24 0 obj 611.8 897.2 734 761.6 666.2 761.6 720.6 544 707.2 734 734 1006 734 734 598.4 272 (answer), Ex 11.2.3 Explain why \(\sum_{n=1}^\infty {3\over n}\) diverges. /Length 465 Proofs for both tests are also given. 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 \(\sum_{n=1}^N {1\over n^4} + 0.005\). >> /Subtype/Type1 Part II. >> We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. A brick wall has 60 bricks in the first row, but each row has 3 fewer bricks than the previous one. Accessibility StatementFor more information contact us atinfo@libretexts.org. Which of the following is the 14th term of the sequence below? /Type/Font Harmonic series and p-series. /Filter /FlateDecode . stream 1. To use integration by parts in Calculus, follow these steps: Decompose the entire integral (including dx) into two factors. Determine whether the following series converge or diverge. (answer), Ex 11.2.2 Explain why \(\sum_{n=1}^\infty {5\over 2^{1/n}+14}\) diverges. >> /Type/Font (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. We also discuss differentiation and integration of power series. stream 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 endstream 15 0 obj 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. Ex 11.10.8 Find the first four terms of the Maclaurin series for \(\tan x\) (up to and including the \( x^3\) term). (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. Images. Quiz 1: 5 questions Practice what you've learned, and level up on the above skills. Here are a set of practice problems for the Series and Sequences chapter of the Calculus II notes. 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. Math 129 - Calculus II. (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. 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). 11.E: Sequences and Series (Exercises) These are homework exercises to accompany David Guichard's "General Calculus" Textmap. Khan Academy is a 501(c)(3) nonprofit organization. 816 816 272 299.2 489.6 489.6 489.6 489.6 489.6 792.7 435.2 489.6 707.2 761.6 489.6 << endobj 979.2 489.6 489.6 489.6] (answer). 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. /Type/Font 1277.8 555.6 1000 1444.4 555.6 1000 1444.4 472.2 472.2 527.8 527.8 527.8 527.8 666.7 489.6 272 489.6 272 272 489.6 544 435.2 544 435.2 299.2 489.6 544 272 299.2 516.8 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). Determine whether the sequence converges or diverges. 5.3.1 Use the divergence test to determine whether a series converges or diverges. Sequences can be thought of as functions whose domain is the set of integers. Don't all infinite series grow to infinity? Other sets by this creator. 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 . Remark. 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 endobj All other trademarks and copyrights are the property of their respective owners. /Filter[/FlateDecode] % /Filter /FlateDecode AP is a registered trademark of the College Board, which has not reviewed this resource. /FontDescriptor 17 0 R Ex 11.1.3 Determine whether \(\{\sqrt{n+47}-\sqrt{n}\}_{n=0}^{\infty}\) converges or diverges. Find the radius and interval of convergence for each series. Strip out the first 3 terms from the series n=1 2n n2 +1 n = 1 2 n n 2 + 1. Which of the following sequences follows this formula. Comparison tests. Root Test In this section we will discuss using the Root Test to determine if an infinite series converges absolutely or diverges. (answer), Ex 11.2.1 Explain why \(\sum_{n=1}^\infty {n^2\over 2n^2+1}\) diverges. Most sections should have a range of difficulty levels in the problems although this will vary from section to section. The Integral Test can be used on a infinite series provided the terms of the series are positive and decreasing. 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. Then click 'Next Question' to answer the next question. endstream endobj 208 0 obj <. The numbers used come from a sequence. Good luck! /Name/F3 Your instructor might use some of these in class. A proof of the Integral Test is also given. (answer), Ex 11.2.4 Compute \(\sum_{n=0}^\infty {4\over (-3)^n}- {3\over 3^n}\). endobj Donate or volunteer today! Our mission is to provide a free, world-class education to anyone, anywhere. A proof of the Root Test is also given. /LastChar 127 You may also use any of these materials for practice. If the series is an alternating series, determine whether it converges absolutely, converges conditionally, or diverges. 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 To log in and use all the features of Khan Academy, please enable JavaScript in your browser. /Subtype/Type1 OR sequences are lists of numbers, where the numbers may or may not be determined by a pattern. 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 . 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] 9 0 obj n a n converges if and only if the integral 1 f ( x) d x converges. All other trademarks and copyrights are the property of their respective owners. We will focus on the basic terminology, limits of sequences and convergence of sequences in this section. >> Integral Test In this section we will discuss using the Integral Test to determine if an infinite series converges or diverges. Integral test. If you're seeing this message, it means we're having trouble loading external resources on our website. 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 % Section 10.3 : Series - Basics. (answer), Ex 11.4.6 Approximate \(\sum_{n=1}^\infty (-1)^{n-1}{1\over n^4}\) to two decimal places. 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 . 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. x[[o6~cX/e`ElRm'1%J$%v)tb]1U2sRV}.l%s\Y UD+q}O+J When you have completed the free practice test, click 'View Results' to see your results. Parametric equations, polar coordinates, and vector-valued functions Calculator-active practice: Parametric equations, polar coordinates, . n = 1 n 2 + 2 n n 3 + 3 n . L7s[AQmT*Z;HK%H0yqt1r8 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. 979.2 489.6 489.6 489.6] 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 stream When given a sum a[n], if the limit as n-->infinity does not exist or does not equal 0, the sum diverges. (answer), Ex 11.2.9 Compute \(\sum_{n=1}^\infty {3^n\over 5^{n+1}}\). Khan Academy is a 501(c)(3) nonprofit organization. 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 /BaseFont/SFGTRF+CMSL12 Ex 11.7.3 Compute \(\lim_{n\to\infty} |a_n|^{1/n}\) for the series \(\sum 1/n^2\). (answer), Ex 11.1.6 Determine whether \(\left\{{2^n\over n! Learn how this is possible, how we can tell whether a series converges, and how we can explore convergence in . 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. If it converges, compute the limit. 252 0 obj <>stream Which of the following sequences follows this formula? After each bounce, the ball reaches a height that is 2/3 of the height from which it previously fell. We will illustrate how we can find a series representation for indefinite integrals that cannot be evaluated by any other method. /BaseFont/VMQJJE+CMR8 For each of the following series, determine which convergence test is the best to use and explain why. )^2\over n^n}\) (answer). Ratio Test In this section we will discuss using the Ratio Test to determine if an infinite series converges absolutely or diverges. xTn0+,ITi](N@ fH2}W"UG'.% Z#>y{!9kJ+ Solution. 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 Which of the following is the 14th term of the sequence below? The sum of the steps forms an innite series, the topic of Section 10.2 and the rest of Chapter 10. 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 4 avwo/MpLv) _C>5p*)i=^m7eE. 722.2 777.8 777.8 611.1 798.5 656.8 526.5 771.4 527.8 718.7 594.9 844.5 544.5 677.8 5.3.2 Use the integral test to determine the convergence of a series. Some infinite series converge to a finite value. These are homework exercises to accompany David Guichard's "General Calculus" Textmap. 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 << %PDF-1.2 Series Infinite geometric series: Series nth-term test: Series Integral test: Series Harmonic series and p-series: Series Comparison tests: . 5.3.3 Estimate the value of a series by finding bounds on its remainder term. In the previous section, we determined the convergence or divergence of several series by . Determine whether each series converges absolutely, converges conditionally, or diverges. Complementary General calculus exercises can be found for other Textmaps and can be accessed here. (answer). /LastChar 127 Absolute and conditional convergence. We will also give many of the basic facts and properties well need as we work with sequences. All rights reserved. /Type/Font Here are a set of practice problems for the Series and Sequences chapter of the Calculus II notes. 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. 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 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! 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. To log in and use all the features of Khan Academy, please enable JavaScript in your browser. MATH 126 Medians and Such. 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 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. In other words, a series is the sum of a sequence. Indiana Core Assessments Mathematics: Test Prep & Study Guide. 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. The book contains eight practice tests five practice tests for Calculus AB and three practice tests for Calculus BC. Question 5 5. 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 . n = 1 n2 + 2n n3 + 3n2 + 1. (answer), Ex 11.9.2 Find a power series representation for \(1/(1-x)^2\). 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 0 stream 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 Use the Comparison Test to determine whether each series in exercises 1 - 13 converges or diverges. Given item A, which of the following would be the value of item B? Which is the infinite sequence starting with 1 where each number is the previous number times 3? Ex 11.1.2 Use the squeeze theorem to show that \(\lim_{n\to\infty} {n!\over n^n}=0\). 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 (answer), Ex 11.11.1 Find a polynomial approximation for \(\cos x\) on \([0,\pi]\), accurate to \( \pm 10^{-3}\) (answer), Ex 11.11.2 How many terms of the series for \(\ln x\) centered at 1 are required so that the guaranteed error on \([1/2,3/2]\) is at most \( 10^{-3}\)? Which of the following sequences is NOT a geometric sequence? Which one of these sequences is a finite sequence? The following is a list of worksheets and other materials related to Math 129 at the UA. (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 \(\sum_{n=2}^N {1\over n(\ln n)^2} + 0.005\). We will examine Geometric Series, Telescoping Series, and Harmonic Series. 70 terms. << 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 . (You may want to use Sage or a similar aid.) Published by Wiley. UcTIjeB#vog-TM'FaTzG(:k-BNQmbj}'?^h<=XgS/]o4Ilv%Jm (answer), Ex 11.10.10 Use a combination of Maclaurin series and algebraic manipulation to find a series centered at zero for \( xe^{-x}\). 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 /Type/Font }\right\}_{n=0}^{\infty}\) converges or diverges. (b) /Length 1247 /FontDescriptor 20 0 R (answer), Ex 11.1.4 Determine whether \(\left\{{n^2+1\over (n+1)^2}\right\}_{n=0}^{\infty}\) converges or diverges. If it converges, compute the limit. 833.3 833.3 833.3 833.3 1444.4 1277.8 555.6 1111.1 1111.1 1111.1 1111.1 1111.1 944.4 489.6 272 489.6 272 272 489.6 544 435.2 544 435.2 299.2 489.6 544 272 299.2 516.8 /FontDescriptor 23 0 R At this time, I do not offer pdf's for solutions to individual problems. /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 We will also see how we can use the first few terms of a power series to approximate a function. )Ltgx?^eaT'&+n+hN4*D^UR!8UY@>LqS%@Cp/-12##DR}miBw6"ja+WjU${IH$5j!j-I1 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. Our mission is to provide a free, world-class education to anyone, anywhere. Choose your answer to the question and click 'Continue' to see how you did. When you have completed the free practice test, click 'View Results' to see your results. When you have completed the free practice test, click 'View Results' to see your results. Which of the following sequences is NOT a geometric sequence? /Name/F6 Ex 11.7.1 Compute \(\lim_{n\to\infty} |a_{n+1}/a_n|\) for the series \(\sum 1/n^2\). Each term is the difference of the previous two terms. 611.8 897.2 734 761.6 666.2 761.6 720.6 544 707.2 734 734 1006 734 734 598.4 272 /Widths[611.8 816 761.6 679.6 652.8 734 707.2 761.6 707.2 761.6 707.2 571.2 544 544 508.8 453.8 482.6 468.9 563.7 334 405.1 509.3 291.7 856.5 584.5 470.7 491.4 434.1 What if the interval is instead \([1,3/2]\)? (answer), Ex 11.2.7 Compute \(\sum_{n=0}^\infty {3^{n+1}\over 7^{n+1}}\). Chapters include Linear Ex 11.9.5 Find a power series representation for \(\int\ln(1-x)\,dx\). 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 666.7 1000 1000 1000 1000 1055.6 1055.6 1055.6 777.8 666.7 666.7 450 450 450 450 18 0 obj My calculus 2 exam on sequence, infinite series & power seriesThe exam: https://bit.ly/36OHYcsAll the convergence tests: https://bit.ly/2IzqokhBest friend an. Good luck! Which is the finite sequence of four multiples of 9, starting with 9? Contact us by phone at (877)266-4919, or by mail at 100ViewStreet#202, MountainView, CA94041. 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. (answer). Calculus 2. Therefore the radius of convergence is R = , and the interval of convergence is ( - , ).

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