1. Comparison tests. 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. Donate or volunteer today! For problems 1 - 3 perform an index shift so that the series starts at n = 3 n = 3. (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. >> To log in and use all the features of Khan Academy, please enable JavaScript in your browser. 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. The Integral Test can be used on a infinite series provided the terms of the series are positive and decreasing. )^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 \(\ln 2\). Worksheets. The sum of the steps forms an innite series, the topic of Section 10.2 and the rest of Chapter 10. /Filter /FlateDecode Note that some sections will have more problems than others and some will have more or less of a variety of problems. 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 We also derive some well known formulas for Taylor series of \({\bf e}^{x}\) , \(\cos(x)\) and \(\sin(x)\) around \(x=0\). (answer). (answer). 777.8 444.4 444.4 444.4 611.1 777.8 777.8 777.8 777.8] Then click 'Next Question' to answer the next question. Each review chapter is packed with equations, formulas, and examples with solutions, so you can study smarter and score a 5! (b) Parametric equations, polar coordinates, and vector-valued functions Calculator-active practice: Parametric equations, polar coordinates, . /Name/F1 << << A proof of the Integral Test is also given. If L = 1, then the test is inconclusive. Which of the following represents the distance the ball bounces from the first to the seventh bounce with sigma notation? 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. 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_. If it converges, compute the limit. 5.3.1 Use the divergence test to determine whether a series converges or diverges. 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 (answer), Ex 11.10.10 Use a combination of Maclaurin series and algebraic manipulation to find a series centered at zero for \( xe^{-x}\). Accessibility StatementFor more information contact us atinfo@libretexts.org. Section 10.3 : Series - Basics. endobj It turns out the answer is no. Good luck! Ex 11.1.3 Determine whether {n + 47 n} . >> Psychological Research & Experimental Design, All Teacher Certification Test Prep Courses. 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 of the sequences below has the recursive rule where each number is the previous number times 2? /Length 465 70 terms. 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 /Filter[/FlateDecode] 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 What is the radius of convergence? At this time, I do not offer pdf's for solutions to individual problems. 68 0 obj 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. 777.8 777.8] endobj /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 stream The LibreTexts libraries arePowered by NICE CXone Expertand are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. /Name/F6 /Subtype/Type1 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. /Length 569 Legal. /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 xTn0+,ITi](N@ fH2}W"UG'.% Z#>y{!9kJ+ ]^e-V!2 F. |: The Ratio Test shows us that regardless of the choice of x, the series converges. Published by Wiley. /Subtype/Type1 Convergence/Divergence of Series In this section we will discuss in greater detail the convergence and divergence of infinite series. (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 \(\sum_{n=0}^N {1\over e^n} + 10^{-4}\). (answer), Ex 11.1.4 Determine whether \(\left\{{n^2+1\over (n+1)^2}\right\}_{n=0}^{\infty}\) converges or diverges. 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 . 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 . 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 Ratio test. /LastChar 127 /FontDescriptor 14 0 R n a n converges if and only if the integral 1 f ( x) d x converges. /FirstChar 0 (answer), Ex 11.4.6 Approximate \(\sum_{n=1}^\infty (-1)^{n-1}{1\over n^4}\) to two decimal places. Level up on all the skills in this unit and collect up to 2000 Mastery points! hbbd```b``~"A$" "Y`L6`RL,-`sA$w64= f[" RLMu;@jAl[`3H^Ne`?$4 1 2 + 1 4 + 1 8 + = n=1 1 2n = 1 We will need to be careful, but it turns out that we can . If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. (answer), Ex 11.2.8 Compute \(\sum_{n=1}^\infty \left({3\over 5}\right)^n\). 17 0 obj A proof of the Ratio Test is also given. /FirstChar 0 Part II. 611.8 897.2 734 761.6 666.2 761.6 720.6 544 707.2 734 734 1006 734 734 598.4 272 /Length 1722 Taylor Series In this section we will discuss how to find the Taylor/Maclaurin Series for a function. Choose the equation below that represents the rule for the nth term of the following geometric sequence: 128, 64, 32, 16, 8, . 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. Alternating series test. A proof of the Alternating Series Test is also given. (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 \(\sum_{n=2}^N {1\over n(\ln n)^2} + 0.005\). 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. Math 129 - Calculus II. (answer), Ex 11.3.11 Find an \(N\) so that \(\sum_{n=1}^\infty {\ln n\over n^2}\) is between \(\sum_{n=1}^N {\ln n\over n^2}\) and \(\sum_{n=1}^N {\ln n\over n^2} + 0.005\). 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. copyright 2003-2023 Study.com. Some infinite series converge to a finite value. Let the factor without dx equal u and the factor with dx equal dv. 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? We also discuss differentiation and integration of power series. Which of the following is the 14th term of the sequence below? %%EOF 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 Determine whether the sequence converges or diverges. Which is the finite sequence of four multiples of 9, starting with 9? /Filter /FlateDecode Images. 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 . 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. 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. % (a) $\sum_{n=1}^{\infty} \frac{(-1)^n}{\sqrt{n}}$ (b) $\sum_{n=1}^{\infty}(-1)^n \frac{n}{2 n-1}$ Math 106 (Calculus II): old exams. 2 6 points 2.
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