calculus 2 series and sequences practice test

Don't all infinite series grow to infinity? /BaseFont/PSJLQR+CMEX10 hbbd```b``~"A$" "Y`L6`RL,-`sA$w64= f[" RLMu;@jAl[`3H^Ne`?$4 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. /Subtype/Type1 Ex 11.9.5 Find a power series representation for $\int\ln(1-x)\,dx$. For each function, find the Maclaurin series or Taylor series centered at $a$, and the radius of convergence. /LastChar 127 The following is a list of worksheets and other materials related to Math 129 at the UA. All other trademarks and copyrights are the property of their respective owners. Choose your answer to the question and click 'Continue' to see how you did. A brick wall has 60 bricks in the first row, but each row has 3 fewer bricks than the previous one. 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 11.4.1 $\sum_{n=1}^\infty {(-1)^{n-1}\over 2n+5}$ (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. Divergence Test. Parametric equations, polar coordinates, and vector-valued functions Calculator-active practice: Parametric equations, polar coordinates, . Determine whether the series is convergent or divergent. S.QBt'(d|/"XH4!qbnEriHX)Gs2qN/G jq8$$< >> (answer), Ex 11.2.6 Compute $\sum_{n=0}^\infty {4^{n+1}\over 5^n}$. 24 0 obj 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. Comparison tests. << Complementary General calculus exercises can be found for other Textmaps and can be accessed here. Ex 11.1.2 Use the squeeze theorem to show that limn n! %|S#?\A@D-oS)lW=??nn}y]Tb!!o_=;]ha,J[. 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 (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. Each term is the sum of the previous two terms. 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. What is the radius of convergence? 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? 611.8 897.2 734 761.6 666.2 761.6 720.6 544 707.2 734 734 1006 734 734 598.4 272 We will also give many of the basic facts and properties well need as we work with sequences. /Length 2492 /Name/F5 Alternating Series Test - In this section we will discuss using the Alternating Series Test to determine if an infinite series converges or diverges. The numbers used come from a sequence. Quiz 1: 5 questions Practice what you've learned, and level up on the above skills. /Subtype/Type1 Ex 11.7.9 Prove theorem 11.7.3, the root test. We will illustrate how we can find a series representation for indefinite integrals that cannot be evaluated by any other method. A review of all series tests. Infinite series are sums of an infinite number of terms. 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 To log in and use all the features of Khan Academy, please enable JavaScript in your browser. Contact us by phone at (877)266-4919, or by mail at 100ViewStreet#202, MountainView, CA94041. (answer), Ex 11.2.9 Compute $\sum_{n=1}^\infty {3^n\over 5^{n+1}}$. /Name/F1 When you have completed the free practice test, click 'View Results' to see your results. 805.6 805.6 1277.8 1277.8 811.1 811.1 875 875 666.7 666.7 666.7 666.7 666.7 666.7 826.4 531.3 958.7 1076.8 826.4 295.1 295.1 531.3 885.4 531.3 885.4 826.4 295.1 413.2 Determine whether the sequence converges or diverges. %PDF-1.5 At this time, I do not offer pdf's for solutions to individual problems. xu? ~k"xPeEV4Vcwww \ a:5d*%30EU9>,e92UU3Voj/$f BS!.eSloaY&h&Urm!U3L%g@'>`|$ogJ /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 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}}}}} $. 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. /BaseFont/CQGOFL+CMSY10 11.E: Sequences and Series (Exercises) These are homework exercises to accompany David Guichard's "General Calculus" Textmap. 0 >> Images. 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 xTn0+,ITi](N@ fH2}W"UG'.% Z#>y{!9kJ+ (answer). After each bounce, the ball reaches a height that is 2/3 of the height from which it previously fell. Learn for free about math, art, computer programming, economics, physics, chemistry, biology, medicine, finance, history, and more. 70 terms. 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 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. % Which of the following is the 14th term of the sequence below? /Name/F2 18 0 obj endstream Premium members get access to this practice exam along with our entire library of lessons taught by subject matter experts. bmkraft7. 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. Then click 'Next Question' to answer the next question. /FontDescriptor 14 0 R Here are a set of practice problems for the Series and Sequences chapter of the Calculus II notes. Integral Test: If a n = f ( n), where f ( x) is a non-negative non-increasing function, then. /Subtype/Type1 endobj A proof of the Root Test is also given. Ex 11.10.8 Find the first four terms of the Maclaurin series for $\tan x$ (up to and including the $ x^3$ term). /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 . 590.3 885.4 885.4 295.1 324.7 531.3 531.3 531.3 531.3 531.3 795.8 472.2 531.3 767.4 531.3 590.3 560.8 414.1 419.1 413.2 590.3 560.8 767.4 560.8 560.8 472.2 531.3 1062.5 In exercises 3 and 4, do not attempt to determine whether the endpoints are in the interval of convergence. 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). Absolute and conditional convergence. Legal. Consider the series n a n. Divergence Test: If lim n a n 0, then n a n diverges. /Type/Font Taylor Series In this section we will discuss how to find the Taylor/Maclaurin Series for a function. What is the 83rd term of the sequence 91, 87, 83, 79, ( = a. 611.8 897.2 734 761.6 666.2 761.6 720.6 544 707.2 734 734 1006 734 734 598.4 272 At this time, I do not offer pdf's for . (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. To use the Geometric Series formula, the function must be able to be put into a specific form, which is often impossible. 21 terms. 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. What if the interval is instead $[1,3/2]$? MULTIPLE CHOICE: Circle the best answer. 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 . Quiz 2: 8 questions Practice what you've learned, and level up on the above skills. ]^e-V!2 F. endobj MATH 126 Medians and Such. However, use of this formula does quickly illustrate how functions can be represented as a power series. (You may want to use Sage or a similar aid.) (b) /LastChar 127 If it con-verges, nd the limit. Solution. About this unit. We will also determine a sequence is bounded below, bounded above and/or bounded. 21 0 obj Series The Basics In this section we will formally define an infinite series. >> 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 11.11.1 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}$? 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. We will also see how we can use the first few terms of a power series to approximate a function. Strip out the first 3 terms from the series n=1 2n n2 +1 n = 1 2 n n 2 + 1. Question 5 5. 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. Remark. stream (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 $\sum_{n=2}^N {1\over n(\ln n)^2} + 0.005$. 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 UcTIjeB#vog-TM'FaTzG(:k-BNQmbj}'?^h<=XgS/]o4Ilv%Jm /FirstChar 0 The Integral Test can be used on a infinite series provided the terms of the series are positive and decreasing. << Here are a set of practice problems for the Series and Sequences chapter of the Calculus II notes. 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 endstream endobj 208 0 obj <. Choose the equation below that represents the rule for the nth term of the following geometric sequence: 128, 64, 32, 16, 8, . 62 0 obj stream Sequences and Numerical series. Donate or volunteer today! 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