8 0 obj 272 761.6 462.4 462.4 761.6 734 693.4 707.2 747.8 666.2 639 768.3 734 353.2 503 761.2 531.3 531.3 531.3 295.1 295.1 295.1 826.4 501.7 501.7 826.4 795.8 752.1 767.4 811.1 Premium members get access to this practice exam along with our entire library of lessons taught by subject matter experts. The Alternating Series Test can be used only if the terms of the Determine whether the series is convergent or divergent. Consider the series n a n. Divergence Test: If lim n a n 0, then n a n diverges. raVQ1CKD3` rO:H\hL[+?zWl'oDpP% bhR5f7RN `1= SJt{p9kp5,W+Y.e7) Zy\BP>+``;qI^%$x=%f0+!.=Q7HgbjfCVws,NL)%"pcS^ {tY}vf~T{oFe{nB\bItw$nku#pehXWn8;ZW]/v_nF787nl{ y/@U581$&DN>+gt (answer), Ex 11.2.1 Explain why \(\sum_{n=1}^\infty {n^2\over 2n^2+1}\) diverges. Which of the following sequences follows this formula. 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 (a) $\sum_{n=1}^{\infty} \frac{(-1)^n}{\sqrt{n}}$ (b) $\sum_{n=1}^{\infty}(-1)^n \frac{n}{2 n-1}$ >> L7s[AQmT*Z;HK%H0yqt1r8 At this time, I do not offer pdf's for solutions to individual problems. 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 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. 15 0 obj copyright 2003-2023 Study.com. Series are sums of multiple terms. Some infinite series converge to a finite value. /Subtype/Type1 endstream 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 MULTIPLE CHOICE: Circle the best answer. Then click 'Next Question' to answer the next question. (answer). stream Below are some general cases in which each test may help: P-Series Test: The series be written in the form: P 1 np Geometric Series Test: When the series can be written in the form: P a nrn1 or P a nrn Direct Comparison Test: When the given series, a 17 0 obj /Name/F3 /Filter[/FlateDecode] Calculus II - Series & Sequences (Practice Problems) - Lamar University (answer), Ex 11.10.9 Use a combination of Maclaurin series and algebraic manipulation to find a series centered at zero for \( x\cos (x^2)\). << endobj 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 . 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 /Filter /FlateDecode stream 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. /LastChar 127 326.4 272 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 272 272 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 . Ex 11.10.8 Find the first four terms of the Maclaurin series for \(\tan x\) (up to and including the \( x^3\) term). 9 0 obj Good luck! Premium members get access to this practice exam along with our entire library of lessons taught by subject matter experts. << 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 n a n converges if and only if the integral 1 f ( x) d x converges. (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 \( 10^{-3}\)? PDF FINAL EXAM CALCULUS 2 - Department of Mathematics Don't all infinite series grow to infinity? Strategies for Testing Series - University of Texas at Austin 26 0 obj hbbd```b``~"A$" "Y`L6`RL,-`sA$w64= f[" RLMu;@jAl[`3H^Ne`?$4 YesNo 2.(b). %%EOF Comparison Test: This applies . We will also determine a sequence is bounded below, bounded above and/or bounded. All other trademarks and copyrights are the property of their respective owners. /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. The practice tests are composed /Length 1247 stream Ex 11.1.2 Use the squeeze theorem to show that \(\lim_{n\to\infty} {n!\over n^n}=0\). Calculus II - Sequences and Series Flashcards | Quizlet Which of the following is the 14th term of the sequence below? We also derive some well known formulas for Taylor series of \({\bf e}^{x}\) , \(\cos(x)\) and \(\sin(x)\) around \(x=0\). 979.2 489.6 489.6 489.6] We will examine Geometric Series, Telescoping Series, and Harmonic Series. 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. Ex 11.1.3 Determine whether \(\{\sqrt{n+47}-\sqrt{n}\}_{n=0}^{\infty}\) converges or diverges. If a geometric series begins with the following term, what would the next term be? Then click 'Next Question' to answer the next question. /LastChar 127 % To use the Geometric Series formula, the function must be able to be put into a specific form, which is often impossible. 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 \(\sum_{n=1}^N {1\over n^4} + 0.005\). More on Sequences In this section we will continue examining sequences. The steps are terms in the sequence. Each term is the sum of the previous two terms. 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? Images. /Subtype/Type1 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. (You may want to use Sage or a similar aid.) 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 . 666.7 1000 1000 1000 1000 1055.6 1055.6 1055.6 777.8 666.7 666.7 450 450 450 450 Chapter 10 : Series and Sequences. Research Methods Midterm. Then click 'Next Question' to answer the next question. endobj Some infinite series converge to a finite value. /FirstChar 0 489.6 272 489.6 272 272 489.6 544 435.2 544 435.2 299.2 489.6 544 272 299.2 516.8 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. We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. We will also give the Divergence Test for series in this section. endobj PDF M 172 - Calculus II - Chapter 10 Sequences and Series /FontDescriptor 8 0 R All rights reserved. Integral Test In this section we will discuss using the Integral Test to determine if an infinite series converges or diverges. endstream /Type/Font 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. Applications of Series In this section we will take a quick look at a couple of applications of series. /Name/F5 << /Name/F2 Use the Comparison Test to determine whether each series in exercises 1 - 13 converges or diverges. 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 Math 106 (Calculus II): old exams | Mathematics | Bates College Accessibility StatementFor more information contact us atinfo@libretexts.org. endstream UcTIjeB#vog-TM'FaTzG(:k-BNQmbj}'?^h<=XgS/]o4Ilv%Jm /Subtype/Type1 The chapter headings refer to Calculus, Sixth Edition by Hughes-Hallett et al. Example 1. Good luck! PDF Practice Problems Series & Sequences - MR. SOLIS' WEEBLY 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 Derivatives, Integrals, Sequences & Series, and Vector Valued Functions. Solving My Calc 2 Exam#3 (Sequence, Infinite Series & Power Series) Psychological Research & Experimental Design, All Teacher Certification Test Prep Courses. 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@ " Divergence Test. (answer), Ex 11.2.7 Compute \(\sum_{n=0}^\infty {3^{n+1}\over 7^{n+1}}\). 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. Ex 11.8.1 \(\sum_{n=0}^\infty n x^n\) (answer), Ex 11.8.2 \(\sum_{n=0}^\infty {x^n\over n! Power Series and Functions In this section we discuss how the formula for a convergent Geometric Series can be used to represent some functions as power series. Learn how this is possible, how we can tell whether a series converges, and how we can explore convergence in Taylor and Maclaurin series. Calculus 2. Series The Basics In this section we will formally define an infinite series. 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 xu? ~k"xPeEV4Vcwww \ a:5d*%30EU9>,e92UU3Voj/$f BS!.eSloaY&h&Urm!U3L%g@'>`|$ogJ If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. Note that some sections will have more problems than others and some will have more or less of a variety of problems. 5.3 The Divergence and Integral Tests - Calculus Volume 2 - OpenStax 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. /Type/Font When given a sum a[n], if the limit as n-->infinity does not exist or does not equal 0, the sum diverges. %PDF-1.5 /Subtype/Type1 Alternating Series Test - In this section we will discuss using the Alternating Series Test to determine if an infinite series converges or diverges. (answer), Ex 11.10.10 Use a combination of Maclaurin series and algebraic manipulation to find a series centered at zero for \( xe^{-x}\). /BaseFont/VMQJJE+CMR8 Series Infinite geometric series: Series nth-term test: Series Integral test: Series Harmonic series and p-series: Series Comparison tests: . In other words, a series is the sum of a sequence. PDF Read Free Answers To Algebra 2 Practice B Ellipses Most sections should have a range of difficulty levels in the problems although this will vary from section to section. 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. If you're seeing this message, it means we're having trouble loading external resources on our website. 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 Alternating series test. Calculus (single and multi-variable) Ordinary Differential equations (upto 2nd order linear with Laplace transforms, including Dirac Delta functions and Fourier Series. /Length 2492 . Ex 11.1.1 Compute \(\lim_{x\to\infty} x^{1/x}\). (answer), Ex 11.2.2 Explain why \(\sum_{n=1}^\infty {5\over 2^{1/n}+14}\) diverges. 12 0 obj Quiz 2: 8 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. In order to use either test the terms of the infinite series must be positive. endobj Series | Calculus 2 | Math | Khan Academy Question 5 5. Calculus II For Dummies Cheat Sheet - dummies << Which is the finite sequence of four multiples of 9, starting with 9? Integral Test: If a n = f ( n), where f ( x) is a non-negative non-increasing function, then. 777.8 444.4 444.4 444.4 611.1 777.8 777.8 777.8 777.8] 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 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. Root Test In this section we will discuss using the Root Test to determine if an infinite series converges absolutely or diverges. (answer). PDF Ap Calculus Ab Bc Kelley Copy - gny.salvationarmy.org A review of all series tests. Good luck! Infinite sequences and series | AP/College Calculus BC - Khan Academy Determine whether the sequence converges or diverges. Each term is the product of the two previous terms. endobj 1 2 + 1 4 + 1 8 + = n=1 1 2n = 1 We will need to be careful, but it turns out that we can . S.QBt'(d|/"XH4!qbnEriHX)Gs2qN/G jq8$$< xTn0+,ITi](N@ fH2}W"UG'.% Z#>y{!9kJ+ { "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]()", "11.07:_Absolute_Convergence" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "11.08:_The_Ratio_and_Root_Tests" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "11.09:_Power_Series" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "11.10:_Calculus_with_Power_Series" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "11.11:_Taylor_Series" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "11.12:_Taylor\'s_Theorem" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "11.13:_Additional_Exercises" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "11.E:_Sequences_and_Series_(Exercises)" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()" }, { "00:_Front_Matter" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "01:_Analytic_Geometry" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "02:_Instantaneous_Rate_of_Change-_The_Derivative" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "03:_Rules_for_Finding_Derivatives" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "04:_Transcendental_Functions" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "05:_Curve_Sketching" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "06:_Applications_of_the_Derivative" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "07:_Integration" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "08:_Techniques_of_Integration" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "09:_Applications_of_Integration" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "10:_Polar_Coordinates_and_Parametric_Equations" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "11:_Sequences_and_Series" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "12:_Three_Dimensions" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "13:_Vector_Functions" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "14:_Partial_Differentiation" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "15:_Multiple_Integration" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "16:_Vector_Calculus" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "17:_Differential_Equations" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "zz:_Back_Matter" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()" }, [ "article:topic", "authorname:guichard", "license:ccbyncsa", "showtoc:no" ], 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), \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}}}\) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\). 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. endstream endobj startxref << (answer), Ex 11.9.3 Find a power series representation for \( 2/(1-x)^3\). /FontDescriptor 11 0 R 1000 1000 777.8 777.8 1000 1000 500 500 1000 1000 1000 777.8 1000 1000 611.1 611.1 /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 %PDF-1.5 Ratio Test In this section we will discuss using the Ratio Test to determine if an infinite series converges absolutely or diverges. ]^e-V!2 F. Ex 11.9.5 Find a power series representation for \(\int\ln(1-x)\,dx\). Maclaurin series of e, sin(x), and cos(x). 5.3.2 Use the integral test to determine the convergence of a series. 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). Each review chapter is packed with equations, formulas, and examples with solutions, so you can study smarter and score a 5! %|S#?\A@D-oS)lW=??nn}y]Tb!!o_=;]ha,J[. Math 129 - Calculus II Worksheets - University of Arizona 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_. 5.3.1 Use the divergence test to determine whether a series converges or diverges. Complementary General calculus exercises can be found for other Textmaps and can be accessed here. (answer), Ex 11.2.4 Compute \(\sum_{n=0}^\infty {4\over (-3)^n}- {3\over 3^n}\). /LastChar 127 777.8 777.8] PDF Arithmetic Sequences And Series Practice Problems /FirstChar 0 Sequences can be thought of as functions whose domain is the set of integers. Indiana Core Assessments Mathematics: Test Prep & Study Guide. A proof of the Ratio Test is also given. 979.2 489.6 489.6 489.6] 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. endobj in calculus coursesincluding Calculus, Calculus II, Calculus III, AP Calculus and Precalculus. Determine whether the series converge or diverge. 9.8 Power Series Chapter 9 Sequences and Series Calculus II /Filter /FlateDecode Sequences In this section we define just what we mean by sequence in a math class and give the basic notation we will use with them. 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. |: The Ratio Test shows us that regardless of the choice of x, the series converges.