(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. /Name/F6 Sequences can be thought of as functions whose domain is the set of integers. (answer), Ex 11.2.3 Explain why \(\sum_{n=1}^\infty {3\over n}\) diverges. 5.3.2 Use the integral test to determine the convergence of a series. Worksheets. 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] ,vEmO8/OuNVRaLPqB.*l. 70 terms. %|S#?\A@D-oS)lW=??nn}y]Tb!!o_=;]ha,J[. Divergence Test. How many bricks are in the 12th row? 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 /Name/F4 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. Each term is the difference of the previous two terms. Ex 11.1.2 Use the squeeze theorem to show that \(\lim_{n\to\infty} {n!\over n^n}=0\). Quiz 1: 5 questions Practice what you've learned, and level up on the above skills. 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 n = 1 n2 + 2n n3 + 3n2 + 1. To use integration by parts in Calculus, follow these steps: Decompose the entire integral (including dx) into two factors. /Type/Font In exercises 3 and 4, do not attempt to determine whether the endpoints are in the interval of convergence. 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 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. 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. Math 106 (Calculus II): old exams. /Filter /FlateDecode Your instructor might use some of these in class. (answer), Ex 11.1.4 Determine whether \(\left\{{n^2+1\over (n+1)^2}\right\}_{n=0}^{\infty}\) converges or diverges. /Type/Font /FirstChar 0 326.4 272 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 272 272 Here are a set of practice problems for the Series and Sequences chapter of the Calculus II notes. 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 Series Infinite geometric series: Series nth-term test: Series Integral test: Series Harmonic series and p-series: Series Comparison tests: . 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. In other words, a series is the sum of a sequence. Which of the following sequences is NOT a geometric sequence? 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 Part II. Which one of these sequences is a finite sequence? 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 (answer), Ex 11.4.6 Approximate \(\sum_{n=1}^\infty (-1)^{n-1}{1\over n^4}\) to two decimal places. Good luck! }\right\}_{n=0}^{\infty}\) converges 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 Premium members get access to this practice exam along with our entire library of lessons taught by subject matter experts. (answer), Ex 11.2.4 Compute \(\sum_{n=0}^\infty {4\over (-3)^n}- {3\over 3^n}\). Khan Academy is a 501(c)(3) nonprofit organization. 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 Alternating Series Test - In this section we will discuss using the Alternating Series Test to determine if an infinite series converges or diverges. Determine whether the series converge or diverge. (1 point) Is the integral Z 1 1 1 x2 dx an improper integral? Calculus II-Sequences and Series. Given item A, which of the following would be the value of item B? /Length 1722 Images. stream 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, \( \displaystyle \sum\limits_{n = 1}^\infty {\left( {n{2^n} - {3^{1 - n}}} \right)} \), \( \displaystyle \sum\limits_{n = 7}^\infty {\frac{{4 - n}}{{{n^2} + 1}}} \), \( \displaystyle \sum\limits_{n = 2}^\infty {\frac{{{{\left( { - 1} \right)}^{n - 3}}\left( {n + 2} \right)}}{{{5^{1 + 2n}}}}} \). Complementary General calculus exercises can be found for other Textmaps and can be accessed here. /FontDescriptor 14 0 R /FontDescriptor 20 0 R The practice tests are composed In the previous section, we determined the convergence or divergence of several series by . Which of the following sequences follows this formula. These are homework exercises to accompany David Guichard's "General Calculus" Textmap. Consider the series n a n. Divergence Test: If lim n a n 0, then n a n diverges. A ball is dropped from an unknown height (h) and it repeatedly bounces on the floor. /Filter /FlateDecode >> 531.3 531.3 531.3 295.1 295.1 295.1 826.4 501.7 501.7 826.4 795.8 752.1 767.4 811.1 It turns out the answer is no. Good luck! 0 Ex 11.1.3 Determine whether \(\{\sqrt{n+47}-\sqrt{n}\}_{n=0}^{\infty}\) converges or diverges. 1000 1000 777.8 777.8 1000 1000 500 500 1000 1000 1000 777.8 1000 1000 611.1 611.1 762 689.7 1200.9 820.5 796.1 695.6 816.7 847.5 605.6 544.6 625.8 612.8 987.8 713.3 If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. (answer), Ex 11.10.10 Use a combination of Maclaurin series and algebraic manipulation to find a series centered at zero for \( xe^{-x}\). (answer), Ex 11.2.6 Compute \(\sum_{n=0}^\infty {4^{n+1}\over 5^n}\). 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, Convergent & divergent geometric series (with manipulation), 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, Integrals & derivatives of functions with known power series, Interval of convergence for derivative and integral, Converting explicit series terms to summation notation, Converting explicit series terms to summation notation (n 2), 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. My calculus 2 exam on sequence, infinite series & power seriesThe exam: https://bit.ly/36OHYcsAll the convergence tests: https://bit.ly/2IzqokhBest friend an. 666.7 1000 1000 1000 1000 1055.6 1055.6 1055.6 777.8 666.7 666.7 450 450 450 450 /FirstChar 0 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. (answer), Ex 11.2.8 Compute \(\sum_{n=1}^\infty \left({3\over 5}\right)^n\). (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 \(\sum_{n=2}^N {1\over n(\ln n)^2} + 0.005\). Determine whether the following series converge or diverge. Series are sums of multiple terms. SAT Practice Questions- All Maths; SAT Practice Test Questions- Reading , Writing and Language; KS 1-2 Math, Science and SAT . xTn0+,ITi](N@ fH2}W"UG'.% Z#>y{!9kJ+ After each bounce, the ball reaches a height that is 2/3 of the height from which it previously fell. Complementary General calculus exercises can be found for other Textmaps and can be accessed here. If it converges, compute the limit. /Name/F1 Derivatives, Integrals, Sequences & Series, and Vector Valued Functions. Section 10.3 : Series - Basics. %PDF-1.2 A proof of the Alternating Series Test is also given. 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). /Name/F3 To log in and use all the features of Khan Academy, please enable JavaScript in your browser. 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 . 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. /Length 2492 Determine whether the series is convergent or divergent. stream Find the radius and interval of convergence for each of the following series: Solution (a) We apply the Ratio Test to the series n = 0 | x n n! 555.6 577.8 577.8 597.2 597.2 736.1 736.1 527.8 527.8 583.3 583.3 583.3 583.3 750 207 0 obj <> endobj Choose the equation below that represents the rule for the nth term of the following geometric sequence: 128, 64, 32, 16, 8, . Good luck! Then click 'Next Question' to answer the next question. Remark. Accessibility StatementFor more information contact us atinfo@libretexts.org. MULTIPLE CHOICE: Circle the best answer. Sequences and Numerical series. Integral Test: If a n = f ( n), where f ( x) is a non-negative non-increasing function, then. 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? 12 0 obj /FirstChar 0 Then click 'Next Question' to answer the . >> 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 Each term is the sum of the previous two terms. 5.3.3 Estimate the value of a series by finding bounds on its remainder term. %%EOF At this time, I do not offer pdf's for . (answer). UcTIjeB#vog-TM'FaTzG(:k-BNQmbj}'?^h<=XgS/]o4Ilv%Jm We also discuss differentiation and integration of power series. 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_. >> L7s[AQmT*Z;HK%H0yqt1r8 /Filter /FlateDecode Published by Wiley. (answer), Ex 11.9.2 Find a power series representation for \(1/(1-x)^2\). Learn how this is possible, how we can tell whether a series converges, and how we can explore convergence in . (b) Determine whether each series converges absolutely, converges conditionally, or diverges. 2 6 points 2. >> At this time, I do not offer pdf's for solutions to individual problems. 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 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 Question 5 5. Choose your answer to the question and click 'Continue' to see how you did. copyright 2003-2023 Study.com. 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 endstream endstream endobj 208 0 obj <. Then we can say that the series diverges without having to do any extra work. 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. endobj The Alternating Series Test can be used only if the terms of the The Alternating Series Test can be used only if the terms of the series alternate in sign. 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. %PDF-1.5 % What is the 83rd term of the sequence 91, 87, 83, 79, ( = a. endobj 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 . << 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. This page titled 11.E: Sequences and Series (Exercises) is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by David Guichard. The sum of the steps forms an innite series, the topic of Section 10.2 and the rest of Chapter 10. 24 0 obj We will illustrate how we can find a series representation for indefinite integrals that cannot be evaluated by any other method. (answer). n a n converges if and only if the integral 1 f ( x) d x converges. /Name/F5 Ex 11.9.5 Find a power series representation for \(\int\ln(1-x)\,dx\). << << 826.4 531.3 958.7 1076.8 826.4 295.1 295.1 531.3 885.4 531.3 885.4 826.4 295.1 413.2 777.8 777.8] 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. Applications of Series In this section we will take a quick look at a couple of applications of series. The Integral Test can be used on a infinite series provided the terms of the series are positive and decreasing. OR sequences are lists of numbers, where the numbers may or may not be determined by a pattern. What is the radius of convergence? >> We will also give many of the basic facts and properties well need as we work with sequences. /LastChar 127 Which of the following sequences follows this formula? For each function, find the Maclaurin series or Taylor series centered at $a$, and the radius of convergence. /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 { "11.01:_Prelude_to_Sequences_and_Series" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "11.02:_Sequences" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "11.03:_Series" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "11.04:_The_Integral_Test" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "11.05:_Alternating_Series" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "11.06:_Comparison_Test" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", 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https://math.libretexts.org/@app/auth/3/login?returnto=https%3A%2F%2Fmath.libretexts.org%2FBookshelves%2FCalculus%2FBook%253A_Calculus_(Guichard)%2F11%253A_Sequences_and_Series%2F11.E%253A_Sequences_and_Series_(Exercises), \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}}}\) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\). /Subtype/Type1 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. We will examine Geometric Series, Telescoping Series, and Harmonic Series. We also derive some well known formulas for Taylor series of \({\bf e}^{x}\) , \(\cos(x)\) and \(\sin(x)\) around \(x=0\). Proofs for both tests are also given. More on Sequences In this section we will continue examining sequences. YesNo 2.(b). Chapters include Linear /LastChar 127 Calculus 2. Ex 11.10.8 Find the first four terms of the Maclaurin series for \(\tan x\) (up to and including the \( x^3\) term). We will also give the Divergence Test for series in this section. 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. All rights reserved. web manual for algebra 2 and pre calculus volume ii pre calculus for dummies jan 20 2021 oers an introduction to the principles of pre calculus covering such topics as functions law of sines and cosines identities sequences series and binomials algebra 2 homework practice workbook oct 29 2021 algebra ii practice tests varsity tutors - Nov 18 . 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. Given that \( \displaystyle \sum\limits_{n = 0}^\infty {\frac{1}{{{n^3} + 1}}} = 1.6865\) determine the value of \( \displaystyle \sum\limits_{n = 2}^\infty {\frac{1}{{{n^3} + 1}}} \). 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. (answer). A brick wall has 60 bricks in the first row, but each row has 3 fewer bricks than the previous one. (answer), Ex 11.2.1 Explain why \(\sum_{n=1}^\infty {n^2\over 2n^2+1}\) diverges. 21 0 obj Choose your answer to the question and click 'Continue' to see how you did. Indiana Core Assessments Mathematics: Test Prep & Study Guide. 777.8 444.4 444.4 444.4 611.1 777.8 777.8 777.8 777.8] /Type/Font /FontDescriptor 8 0 R Khan Academy is a nonprofit with the mission of providing a free, world-class education for anyone, anywhere. . A proof of the Integral Test is also given. MATH 126 Medians and Such. 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. Learning Objectives. 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. 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 11.11.5 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. )^2\over n^n}\) (answer). 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 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. (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 \(\sum_{n=1}^N {\ln n\over n^2} + 0.005\). Which of the following is the 14th term of the sequence below? If a geometric series begins with the following term, what would the next term be? 1111.1 472.2 555.6 1111.1 1511.1 1111.1 1511.1 1111.1 1511.1 1055.6 944.4 472.2 833.3 508.8 453.8 482.6 468.9 563.7 334 405.1 509.3 291.7 856.5 584.5 470.7 491.4 434.1 1) \(\displaystyle \sum^_{n=1}a_n\) where \(a_n=\dfrac{2}{n . A proof of the Ratio Test is 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 \(\sum_{n=1}^N {1\over n^4} + 0.005\). 2.(a). 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 We will determine if a sequence in an increasing sequence or a decreasing sequence and hence if it is a monotonic sequence. At this time, I do not offer pdfs for solutions to individual problems. 772.4 811.3 431.9 541.2 833 666.2 947.3 784.1 748.3 631.1 775.5 745.3 602.2 573.9 x=S0 What is the sum of all the even integers from 2 to 250? >> 979.2 489.6 489.6 489.6] 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 To log in and use all the features of Khan Academy, please enable JavaScript in your browser. /Filter /FlateDecode 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. 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 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. /LastChar 127 Level up on all the skills in this unit and collect up to 2000 Mastery points! Research Methods Midterm. 17 0 obj Comparison tests. |: The Ratio Test shows us that regardless of the choice of x, the series converges. /BaseFont/CQGOFL+CMSY10 )Ltgx?^eaT'&+n+hN4*D^UR!8UY@>LqS%@Cp/-12##DR}miBw6"ja+WjU${IH$5j!j-I1 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 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 rule represents the nth term in the sequence 9, 16, 23, 30? Ex 11.1.1 Compute \(\lim_{x\to\infty} x^{1/x}\). (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 \( 10^{-3}\)? /FirstChar 0 Comparison Test/Limit Comparison Test In this section we will discuss using the Comparison Test and Limit Comparison Tests to determine if an infinite series converges or diverges.

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