>> When you have completed the free practice test, click 'View Results' to see your results. (answer), Ex 11.4.6 Approximate \(\sum_{n=1}^\infty (-1)^{n-1}{1\over n^4}\) to two decimal places. Complementary General calculus exercises can be found for other Textmaps and can be accessed here. Which of the following is the 14th term of the sequence below? /BaseFont/CQGOFL+CMSY10 /Filter /FlateDecode /Name/F4 /LastChar 127 883.8 992.6 761.6 272 272 489.6 816 489.6 816 761.6 272 380.8 380.8 489.6 761.6 272 << Determine whether the following series converge or diverge. Harmonic series and p-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. /LastChar 127 Which of the following sequences is NOT a geometric sequence? /FontDescriptor 8 0 R Donate or volunteer today! endstream endobj 208 0 obj <. 611.8 897.2 734 761.6 666.2 761.6 720.6 544 707.2 734 734 1006 734 734 598.4 272 4 avwo/MpLv) _C>5p*)i=^m7eE. /FirstChar 0 762 689.7 1200.9 820.5 796.1 695.6 816.7 847.5 605.6 544.6 625.8 612.8 987.8 713.3 Then click 'Next Question' to answer the . The chapter headings refer to Calculus, Sixth Edition by Hughes-Hallett et al. MULTIPLE CHOICE: Circle the best answer. /Subtype/Type1 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_. (answer), Ex 11.10.9 Use a combination of Maclaurin series and algebraic manipulation to find a series centered at zero for \( x\cos (x^2)\). stream (a) $\sum_{n=1}^{\infty} \frac{(-1)^n}{\sqrt{n}}$ (b) $\sum_{n=1}^{\infty}(-1)^n \frac{n}{2 n-1}$ 1 2, 1 4, 1 8, Sequences of values of this type is the topic of this rst section. 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 nn = 0. Then click 'Next Question' to answer the next question. 207 0 obj <> endobj The numbers used come from a sequence. xu? ~k"xPeEV4Vcwww \ a:5d*%30EU9>,e92UU3Voj/$f BS!.eSloaY&h&Urm!U3L%g@'>`|$ogJ /FontDescriptor 14 0 R (b) Which one of these sequences is a finite sequence? 21 0 obj Premium members get access to this practice exam along with our entire library of lessons taught by subject matter experts. 11.E: Sequences and Series (Exercises) These are homework exercises to accompany David Guichard's "General Calculus" Textmap. Free Practice Test Instructions: Choose your answer to the question and click 'Continue' to see how you did. Other sets by this creator. Ex 11.1.1 Compute \(\lim_{x\to\infty} x^{1/x}\). More on Sequences In this section we will continue examining sequences. Find the radius and interval of convergence for each series. << (answer). 979.2 489.6 489.6 489.6] Strategy for Series In this section we give a general set of guidelines for determining which test to use in determining if an infinite series will converge or diverge. Each term is the difference of the previous two terms. Strip out the first 3 terms from the series n=1 2n n2 +1 n = 1 2 n n 2 + 1. How many bricks are in the 12th row? 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 . To use the Geometric Series formula, the function must be able to be put into a specific form, which is often impossible. Alternating series test. /Type/Font 1) \(\displaystyle \sum^_{n=1}a_n\) where \(a_n=\dfrac{2}{n . 12 0 obj n a n converges if and only if the integral 1 f ( x) d x converges. Taylor Series In this section we will discuss how to find the Taylor/Maclaurin Series for a function. /BaseFont/VMQJJE+CMR8 Determine whether the series converge or diverge. 489.6 272 489.6 272 272 489.6 544 435.2 544 435.2 299.2 489.6 544 272 299.2 516.8 722.6 693.1 833.5 795.8 382.6 545.5 825.4 663.6 972.9 795.8 826.4 722.6 826.4 781.6 %|S#?\A@D-oS)lW=??nn}y]Tb!!o_=;]ha,J[. In other words, a series is the sum of a sequence. We will also determine a sequence is bounded below, bounded above and/or bounded. Complementary General calculus exercises can be found for other Textmaps and can be accessed here. A proof of the Integral Test is also given. Given that \( \displaystyle \sum\limits_{n = 0}^\infty {\frac{1}{{{n^3} + 1}}} = 1.6865\) determine the value of \( \displaystyle \sum\limits_{n = 2}^\infty {\frac{1}{{{n^3} + 1}}} \). /BaseFont/BPHBTR+CMMI12 (answer), Ex 11.2.6 Compute \(\sum_{n=0}^\infty {4^{n+1}\over 5^n}\). /Subtype/Type1 Comparison Test: This applies . Consider the series n a n. Divergence Test: If lim n a n 0, then n a n diverges. >> )Ltgx?^eaT'&+n+hN4*D^UR!8UY@>LqS%@Cp/-12##DR}miBw6"ja+WjU${IH$5j!j-I1 The Alternating Series Test can be used only if the terms of the series alternate in sign. Alternating Series Test - In this section we will discuss using the Alternating Series Test to determine if an infinite series converges or diverges. If you're seeing this message, it means we're having trouble loading external resources on our website. 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 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 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. 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. xTn0+,ITi](N@ fH2}W"UG'.% Z#>y{!9kJ+ 777.8 777.8] /Type/Font &/ r Sequences & Series in Calculus Chapter Exam. Published by Wiley. (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 \(\sum_{n=0}^N {1\over e^n} + 10^{-4}\). Ex 11.7.3 Compute \(\lim_{n\to\infty} |a_n|^{1/n}\) for the series \(\sum 1/n^2\). Good luck! << Applications of Series In this section we will take a quick look at a couple of applications of series. 9 0 obj Then click 'Next Question' to answer the next question. All rights reserved. stream Divergence Test. /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 >> For problems 1 3 perform an index shift so that the series starts at \(n = 3\). 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 \(\sum_{n=1}^N {1\over n^4} + 0.005\). << /BaseFont/SFGTRF+CMSL12 What is the sum of all the even integers from 2 to 250? In order to use either test the terms of the infinite series must be positive. stream Which of the following sequences follows this formula? Ex 11.7.9 Prove theorem 11.7.3, the root test. /Type/Font 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 . We will also give many of the basic facts, properties and ways we can use to manipulate a series. /Length 569 For each function, find the Maclaurin series or Taylor series centered at $a$, and the radius of convergence. 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 For each of the following series, determine which convergence test is the best to use and explain why. Ex 11.1.3 Determine whether {n + 47 n} . (answer). /Subtype/Type1 The steps are terms in the sequence. Choose your answer to the question and click 'Continue' to see how you did. /Type/Font /FirstChar 0 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. endobj The practice tests are composed Our mission is to provide a free, world-class education to anyone, anywhere. 0 Here are a set of practice problems for the Series and Sequences chapter of the Calculus II notes. In the previous section, we determined the convergence or divergence of several series by . At this time, I do not offer pdf's for . endobj 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. Good luck! %PDF-1.5 % 413.2 531.3 826.4 295.1 354.2 295.1 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 507.9 433.7 395.4 427.7 483.1 456.3 346.1 563.7 571.2 589.1 483.8 427.7 555.4 505 Integral Test: If a n = f ( n), where f ( x) is a non-negative non-increasing function, then. ,vEmO8/OuNVRaLPqB.*l. 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. 750 750 750 1044.4 1044.4 791.7 791.7 583.3 583.3 638.9 638.9 638.9 638.9 805.6 805.6 326.4 272 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 272 272 copyright 2003-2023 Study.com. (answer), Ex 11.9.2 Find a power series representation for \(1/(1-x)^2\). x=S0 x[[o6~cX/e`ElRm'1%J$%v)tb]1U2sRV}.l%s\Y UD+q}O+J 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. /LastChar 127 endobj }\) (answer), Ex 11.8.3 \(\sum_{n=1}^\infty {n!\over n^n}x^n\) (answer), Ex 11.8.4 \(\sum_{n=1}^\infty {n!\over n^n}(x-2)^n\) (answer), Ex 11.8.5 \(\sum_{n=1}^\infty {(n! 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
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