calculus 2 series and sequences practice test

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. 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. bmkraft7. (answer), Ex 11.2.8 Compute $\sum_{n=1}^\infty \left({3\over 5}\right)^n$. /Length 465 Calculus II-Sequences and Series. x[[o6~cX/e`ElRm'1%J$%v)tb]1U2sRV}.l%s\Y UD+q}O+J Ratio test. Research Methods Midterm. Integral Test In this section we will discuss using the Integral Test to determine if an infinite series converges or diverges. The book contains eight practice tests five practice tests for Calculus AB and three practice tests for Calculus BC. /Type/Font Each term is the sum of the previous two terms. Learn how this is possible, how we can tell whether a series converges, and how we can explore convergence in . To log in and use all the features of Khan Academy, please enable JavaScript in your browser. PDF Calc II: Practice Final Exam - Columbia University PDF Calculus II Series - Things to Consider - California State University 441.3 461.2 353.6 557.3 473.4 699.9 556.4 477.4 454.9 312.5 377.9 623.4 489.6 272] /Subtype/Type1 endobj Alternating series test - Wikipedia 5.3 The Divergence and Integral Tests - Calculus Volume 2 - OpenStax We will also give many of the basic facts, properties and ways we can use to manipulate a series. Determine whether the series converge or diverge. If you . Series are sums of multiple terms. The practice tests are composed endstream endobj 208 0 obj <. 826.4 531.3 958.7 1076.8 826.4 295.1 295.1 531.3 885.4 531.3 885.4 826.4 295.1 413.2 777.8 777.8] Calculus (single and multi-variable) Ordinary Differential equations (upto 2nd order linear with Laplace transforms, including Dirac Delta functions and Fourier Series. To use the Geometric Series formula, the function must be able to be put into a specific form, which is often impossible. SAT Practice Questions- All Maths; SAT Practice Test Questions- Reading , Writing and Language; KS 1-2 Math, Science and SAT . In mathematical analysis, the alternating series test is the method used to show that an alternating series is convergent when its terms (1) decrease in absolute value, and (2) approach zero in the limit. hb```9B 7N0$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@ " Comparison Test: This applies . Our mission is to provide a free, world-class education to anyone, anywhere. We will also determine a sequence is bounded below, bounded above and/or bounded. Choose the equation below that represents the rule for the nth term of the following geometric sequence: 128, 64, 32, 16, 8, . Math Journey: Calculus, ODEs, Linear Algebra and Beyond \ _* %l~G"tytO(J*l+X@ uE: m/ ~&Q24Nss(7F!ky=4 Mijo8t;v ZrNRG{I~(iw%0W5b)8*^ yyCCy~Cg{C&BPsTxp%p You may also use any of these materials for practice. Chapter 10 : Series and Sequences. /Length 569 Infinite series are sums of an infinite number of terms. 18 0 obj /FirstChar 0 /FirstChar 0 Some infinite series converge to a finite value. Harmonic series and p-series. 666.7 1000 1000 1000 1000 1055.6 1055.6 1055.6 777.8 666.7 666.7 450 450 450 450 More on Sequences In this section we will continue examining sequences. Khan Academy is a 501(c)(3) nonprofit organization. (answer), Ex 11.1.6 Determine whether $\left\{{2^n\over n! /Filter /FlateDecode 68 0 obj Contact us by phone at (877)266-4919, or by mail at 100ViewStreet#202, MountainView, CA94041. 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 Don't all infinite series grow to infinity? (answer), Ex 11.2.3 Explain why \(\sum_{n=1}^\infty {3\over n}$ diverges. We will illustrate how partial sums are used to determine if an infinite series converges or diverges. 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). (answer). 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. Learn for free about math, art, computer programming, economics, physics, chemistry, biology, medicine, finance, history, and more. Applications of Series In this section we will take a quick look at a couple of applications of series. For each function, find the Maclaurin series or Taylor series centered at $a$, and the radius of convergence. (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. /FontDescriptor 23 0 R These are homework exercises to accompany David Guichard's "General Calculus" Textmap. 5.3.2 Use the integral test to determine the convergence of a series. Calculus II - Sequences and Series Flashcards | Quizlet Most sections should have a range of difficulty levels in the problems although this will vary from section to section. /FontDescriptor 17 0 R endstream /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 Ex 11.1.2 Use the squeeze theorem to show that $\lim_{n\to\infty} {n!\over n^n}=0$. PDF FINAL EXAM CALCULUS 2 - Department of Mathematics (answer), Ex 11.9.4 Find a power series representation for $ 1/(1-x)^3$. 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 numbers used come from a sequence. 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}}}}} $. stream Ex 11.7.2 Compute $\lim_{n\to\infty} |a_{n+1}/a_n|$ for the series $\sum 1/n$. >> A review of all series tests. Then click 'Next Question' to answer the next question. x=S0 Remark. 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}}} $. Legal. 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. /Type/Font In the previous section, we determined the convergence or divergence of several series by . /Widths[663.6 885.4 826.4 736.8 708.3 795.8 767.4 826.4 767.4 826.4 767.4 619.8 590.3 << 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. 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! copyright 2003-2023 Study.com. endobj >> Math 1242: Calculus II - University of North Carolina at Charlotte YesNo 2.(b). Which of the sequences below has the recursive rule where each number is the previous number times 2? 979.2 489.6 489.6 489.6] (You may want to use Sage or a similar aid.) stream Level up on all the skills in this unit and collect up to 2000 Mastery points! Determine whether the sequence converges or diverges. However, use of this formula does quickly illustrate how functions can be represented as a power series. In order to use either test the terms of the infinite series must be positive. Sequences and Series. Final: all from 02/05 and 03/11 exams (except work, separation of variables, and probability) plus sequences, series, convergence tests, power series, Taylor series. Math 106 (Calculus II): old exams. /BaseFont/BPHBTR+CMMI12 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 Sequences can be thought of as functions whose domain is the set of integers. We will also see how we can use the first few terms of a power series to approximate a function. 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. Worksheets. Other sets by this creator. endobj Good luck! 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 A Lot of Series Test Practice Problems - YouTube Accessibility StatementFor more information contact us [email protected]. 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 (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}$. % Sequences and Series: Comparison Test; Taylor Polynomials Practice; Power Series Practice; Calculus II Arc Length of Parametric Equations; 3 Dimensional Lines; Vectors Practice; Meanvariance SD - Mean Variance; Preview text. << endobj xWKoFWlojCpP NDED$(lq"g|3g6X_&F1BXIM5d gOwaN9c,r|9 To use integration by parts in Calculus, follow these steps: Decompose the entire integral (including dx) into two factors. Solution. The chapter headings refer to Calculus, Sixth Edition by Hughes-Hallett et al. 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 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. stream (answer). 15 0 obj }\right\}_{n=0}^{\infty}\) converges or diverges. /Length 2492 Level up on all the skills in this unit and collect up to 2000 Mastery points! 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. 207 0 obj <> endobj 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. 1 2, 1 4, 1 8, Sequences of values of this type is the topic of this rst section. >> Note that some sections will have more problems than others and some will have more or less of a variety of problems. Ex 11.7.5 $\sum_{n=0}^\infty (-1)^{n}{3^n\over 5^n}$ (answer), Ex 11.7.6 $\sum_{n=1}^\infty {n!\over n^n}$ (answer), Ex 11.7.7 $\sum_{n=1}^\infty {n^5\over n^n}$ (answer), Ex 11.7.8 \(\sum_{n=1}^\infty {(n! { "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 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