18 0 obj PDF Arithmetic Sequences And Series Practice Problems /Subtype/Type1 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. MULTIPLE CHOICE: Circle the best answer. 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. Determine whether the sequence converges or diverges. Which of the following sequences is NOT a geometric sequence? Other sets by this creator. 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 Convergence/Divergence of Series In this section we will discuss in greater detail the convergence and divergence of infinite series. Ex 11.7.2 Compute \(\lim_{n\to\infty} |a_{n+1}/a_n|\) for the series \(\sum 1/n\). << ZrNRG{I~(iw%0W5b)8*^ yyCCy~Cg{C&BPsTxp%p If you're seeing this message, it means we're having trouble loading external resources on our website. Then click 'Next Question' to answer the next question. 17 0 obj If it converges, compute the limit. Series are sums of multiple terms. Contact us by phone at (877)266-4919, or by mail at 100ViewStreet#202, MountainView, CA94041. %%EOF 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 PDF M 172 - Calculus II - Chapter 10 Sequences and Series /Filter /FlateDecode We also derive some well known formulas for Taylor series of \({\bf e}^{x}\) , \(\cos(x)\) and \(\sin(x)\) around \(x=0\). The test was used by Gottfried Leibniz and is sometimes known as Leibniz's test, Leibniz's rule, or the Leibniz criterion.The test is only sufficient, not necessary, so some convergent . Mediansandsuch - Medians - MATH 126 Medians and Such Let X be - Studocu /Name/F1 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 /LastChar 127 Calculus II - Series - The Basics (Practice Problems) - Lamar University (answer), Ex 11.9.4 Find a power series representation for \( 1/(1-x)^3\). If L = 1, then the test is inconclusive. Sequences & Series in Calculus Chapter Exam - Study.com /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 Comparison tests. (answer), Ex 11.2.9 Compute \(\sum_{n=1}^\infty {3^n\over 5^{n+1}}\). 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 Worksheets. We will also give many of the basic facts, properties and ways we can use to manipulate a series. After each bounce, the ball reaches a height that is 2/3 of the height from which it previously fell. /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 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). It turns out the answer is no. 326.4 272 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 272 272 (answer), Ex 11.2.3 Explain why \(\sum_{n=1}^\infty {3\over n}\) diverges. /FontDescriptor 11 0 R Math 129 - Calculus II Worksheets - University of Arizona /BaseFont/PSJLQR+CMEX10 Math 106 (Calculus II): old exams. << 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. 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. Sequences & Series in Calculus Chapter Exam. Complementary General calculus exercises can be found for other Textmaps and can be accessed here. Proofs for both tests are also given. /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 (answer), Ex 11.9.2 Find a power series representation for \(1/(1-x)^2\). >> 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. (1 point) Is the integral Z 1 1 1 x2 dx an improper integral? For problems 1 - 3 perform an index shift so that the series starts at n = 3 n = 3. Then determine if the series converges or diverges. (answer), Ex 11.4.6 Approximate \(\sum_{n=1}^\infty (-1)^{n-1}{1\over n^4}\) to two decimal places. /Filter[/FlateDecode] copyright 2003-2023 Study.com. 9.8 Power Series Chapter 9 Sequences and Series Calculus II 750 750 750 1044.4 1044.4 791.7 791.7 583.3 583.3 638.9 638.9 638.9 638.9 805.6 805.6 /FirstChar 0 endobj 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. 5.3.1 Use the divergence test to determine whether a series converges or diverges. Which of the following is the 14th term of the sequence below? Consider the series n a n. Divergence Test: If lim n a n 0, then n a n diverges. /Subtype/Type1 Derivatives, Integrals, Sequences & Series, and Vector Valued Functions. Khan Academy is a 501(c)(3) nonprofit organization. 9 0 obj S.QBt'(d|/"XH4!qbnEriHX)Gs2qN/G jq8$$< endobj Psychological Research & Experimental Design, All Teacher Certification Test Prep Courses. To use the Geometric Series formula, the function must be able to be put into a specific form, which is often impossible. 62 0 obj When you have completed the free practice test, click 'View Results' to see your results. 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 . The chapter headings refer to Calculus, Sixth Edition by Hughes-Hallett et al. (answer), Ex 11.2.7 Compute \(\sum_{n=0}^\infty {3^{n+1}\over 7^{n+1}}\). When you have completed the free practice test, click 'View Results' to see your results. 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Quiz 1: 5 questions Practice what you've learned, and level up on the above skills. &/ r /Name/F3 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 Each term is the difference of the previous two terms. Choose your answer to the question and click 'Continue' to see how you did. << /LastChar 127 Which is the infinite sequence starting with 1 where each number is the previous number times 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, \( \displaystyle \sum\limits_{n = 1}^\infty {\left( {n{2^n} - {3^{1 - n}}} \right)} \), \( \displaystyle \sum\limits_{n = 7}^\infty {\frac{{4 - n}}{{{n^2} + 1}}} \), \( \displaystyle \sum\limits_{n = 2}^\infty {\frac{{{{\left( { - 1} \right)}^{n - 3}}\left( {n + 2} \right)}}{{{5^{1 + 2n}}}}} \). /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 At this time, I do not offer pdfs for solutions to individual problems. All rights reserved. (You may want to use Sage or a similar aid.) L7s[AQmT*Z;HK%H0yqt1r8 /Type/Font 4 avwo/MpLv) _C>5p*)i=^m7eE. xYKs6W(MCG:9iIO=(lkFRI$x$AMN/" J?~i~d cXf9o/r.&Lxy%/D-Yt+"LX]Sfp]Xl-aM_[6(*~mQbh*28AjZx0 =||. endstream 252 0 obj <>stream 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. ]^e-V!2 F. (answer). /Filter /FlateDecode Alternating Series Test In this section we will discuss using the Alternating Series Test to determine if an infinite series converges or diverges. xTn0+,ITi](N@ fH2}W"UG'.% Z#>y{!9kJ+ 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 Ex 11.1.1 Compute \(\lim_{x\to\infty} x^{1/x}\). (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 \(\sum_{n=0}^N {1\over e^n} + 10^{-4}\). The sum of the steps forms an innite series, the topic of Section 10.2 and the rest of Chapter 10. We will also give the Divergence Test for series in this section. My calculus 2 exam on sequence, infinite series & power seriesThe exam: https://bit.ly/36OHYcsAll the convergence tests: https://bit.ly/2IzqokhBest friend an. 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 %PDF-1.2 (answer), Ex 11.2.6 Compute \(\sum_{n=0}^\infty {4^{n+1}\over 5^n}\). Series Infinite geometric series: Series nth-term test: Series Integral test: Series Harmonic series and p-series: Series Comparison tests: . ,vEmO8/OuNVRaLPqB.*l. At this time, I do not offer pdf's for . A Lot of Series Test Practice Problems - YouTube These are homework exercises to accompany David Guichard's "General Calculus" Textmap. 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. We will also see how we can use the first few terms of a power series to approximate a function. UcTIjeB#vog-TM'FaTzG(:k-BNQmbj}'?^h<=XgS/]o4Ilv%Jm Note that some sections will have more problems than others and some will have more or less of a variety of problems. We will illustrate how we can find a series representation for indefinite integrals that cannot be evaluated by any other method. Sequences and Series. AP Calculus AB and BC: Chapter 9 -Infinite Sequences and Series : 9.4 11.E: Sequences and Series (Exercises) These are homework exercises to accompany David Guichard's "General Calculus" Textmap. 666.7 1000 1000 1000 1000 1055.6 1055.6 1055.6 777.8 666.7 666.7 450 450 450 450 %PDF-1.5 % /FontDescriptor 23 0 R 26 0 obj /BaseFont/CQGOFL+CMSY10 /Length 1722 Then we can say that the series diverges without having to do any extra work. 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. (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. It turns out the answer is no. stream 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 Published by Wiley. For each function, find the Maclaurin series or Taylor series centered at $a$, and the radius of convergence. Alternating Series Test - In this section we will discuss using the Alternating Series Test to determine if an infinite series converges or diverges. How many bricks are in the 12th row? If a geometric series begins with the following term, what would the next term be? Series are sums of multiple terms. May 3rd, 2018 - Sequences and Series Practice Test Determine if the sequence is arithmetic Find the term named in the problem 27 4 8 16 Sequences and Series Practice for Test Mr C Miller April 30th, 2018 - Determine if the sequence is arithmetic or geometric the problem 3 Sequences and Series Practice for Test Series Algebra II Math Khan Academy (a) $\sum_{n=1}^{\infty} \frac{(-1)^n}{\sqrt{n}}$ (b) $\sum_{n=1}^{\infty}(-1)^n \frac{n}{2 n-1}$ 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 /FontDescriptor 17 0 R 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 What is the 83rd term of the sequence 91, 87, 83, 79, ( = a. Let the factor without dx equal u and the factor with dx equal dv. Images. 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 . 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. If it con-verges, nd the limit. 70 terms. You may also use any of these materials for practice. !A1axw)}p]WgxmkFftu Good luck! )^2\over n^n}\) (answer). Sequences and Series for Calculus Chapter Exam - Study.com 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 Harmonic series and p-series. Which is the finite sequence of four multiples of 9, starting with 9? )Ltgx?^eaT'&+n+hN4*D^UR!8UY@>LqS%@Cp/-12##DR}miBw6"ja+WjU${IH$5j!j-I1 31 terms. 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. 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 If the series is an alternating series, determine whether it converges absolutely, converges conditionally, or diverges. What if the interval is instead \([1,3/2]\)? nth-term test. Here are a set of practice problems for the Series and Sequences chapter of the Calculus II notes. Then click 'Next Question' to answer the . /LastChar 127 endobj PDF Read Free Answers To Algebra 2 Practice B Ellipses /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 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_. Determine whether the series is convergent or divergent. We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. Math 106 (Calculus II): old exams | Mathematics | Bates College About this unit. Some infinite series converge to a finite value. Ex 11.7.1 Compute \(\lim_{n\to\infty} |a_{n+1}/a_n|\) for the series \(\sum 1/n^2\). 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 Chapters include Linear hbbd```b``~"A$" "Y`L6`RL,-`sA$w64= f[" RLMu;@jAl[`3H^Ne`?$4 /Length 569 << 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 If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. /Type/Font /Subtype/Type1 Ex 11.7.4 Compute \(\lim_{n\to\infty} |a_n|^{1/n}\) for the series \(\sum 1/n\). The book contains eight practice tests five practice tests for Calculus AB and three practice tests for Calculus BC. 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 For each of the following series, determine which convergence test is the best to use and explain why. Alternating series test. /Length 200 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. Sequences and Numerical series. We will examine Geometric Series, Telescoping Series, and Harmonic Series. (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. Solution. When you have completed the free practice test, click 'View Results' to see your results. /FirstChar 0 Which of the following sequences follows this formula? endobj Which rule represents the nth term in the sequence 9, 16, 23, 30? /FirstChar 0 The following is a list of worksheets and other materials related to Math 129 at the UA. /Length 465 777.8 444.4 444.4 444.4 611.1 777.8 777.8 777.8 777.8] 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 . 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 2.(a). Determine whether the following series converge or diverge.