calculus 2 series and sequences practice test

It turns out the answer is no. 772.4 811.3 431.9 541.2 833 666.2 947.3 784.1 748.3 631.1 775.5 745.3 602.2 573.9 531.3 590.3 560.8 414.1 419.1 413.2 590.3 560.8 767.4 560.8 560.8 472.2 531.3 1062.5 238 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 All other trademarks and copyrights are the property of their respective owners. Integral Test: If a n = f ( n), where f ( x) is a non-negative non-increasing function, then. (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. Our mission is to provide a free, world-class education to anyone, anywhere. (answer). /LastChar 127 /LastChar 127 (answer), Ex 11.9.3 Find a power series representation for \( 2/(1-x)^3\). endobj endobj A brick wall has 60 bricks in the first row, but each row has 3 fewer bricks than the previous one. Ex 11.1.3 Determine whether {n + 47 n} . /FontDescriptor 14 0 R (answer). 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 >> 2 6 points 2. /FirstChar 0 |: The Ratio Test shows us that regardless of the choice of x, the series converges. Determine whether the sequence converges or diverges. 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. (answer), Ex 11.10.10 Use a combination of Maclaurin series and algebraic manipulation to find a series centered at zero for \( xe^{-x}\). For each function, find the Maclaurin series or Taylor series centered at $a$, and the radius of convergence. How many bricks are in the 12th row? /Length 569 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. /FirstChar 0 /Subtype/Type1 Good luck! 68 0 obj Level up on all the skills in this unit and collect up to 2000 Mastery points! stream Which of the following sequences follows this formula. (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 \( 10^{-3}\)? When you have completed the free practice test, click 'View Results' to see your results. 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 62 0 obj 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. 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. If it converges, compute the limit. stream The sum of the steps forms an innite series, the topic of Section 10.2 and the rest of Chapter 10. 1000 1000 777.8 777.8 1000 1000 500 500 1000 1000 1000 777.8 1000 1000 611.1 611.1 Here are a set of practice problems for the Series and Sequences chapter of the Calculus II notes. endobj The Integral Test can be used on a infinite series provided the terms of the series are positive and decreasing. Bottom line -- series are just a lot of numbers added together. << To use integration by parts in Calculus, follow these steps: Decompose the entire integral (including dx) into two factors. (answer), Ex 11.2.1 Explain why \(\sum_{n=1}^\infty {n^2\over 2n^2+1}\) diverges. Learn for free about math, art, computer programming, economics, physics, chemistry, biology, medicine, finance, history, and more. /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 /Filter[/FlateDecode] Choose your answer to the question and click 'Continue' to see how you did. 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. Sequences & Series in Calculus Chapter Exam. The chapter headings refer to Calculus, Sixth Edition by Hughes-Hallett et al. More on Sequences In this section we will continue examining sequences. xTn0+,ITi](N@ fH2}W"UG'.% Z#>y{!9kJ+ /FontDescriptor 17 0 R 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 You may also use any of these materials for practice. >> 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. endstream /BaseFont/PSJLQR+CMEX10 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 OR sequences are lists of numbers, where the numbers may or may not be determined by a pattern. 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 xu? ~k"xPeEV4Vcwww \ a:5d*%30EU9>,e92UU3Voj/$f BS!.eSloaY&h&Urm!U3L%g@'>`|$ogJ 11.E: Sequences and Series (Exercises) These are homework exercises to accompany David Guichard's "General Calculus" Textmap. copyright 2003-2023 Study.com. Martha_Austin Teacher. 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 We will determine if a sequence in an increasing sequence or a decreasing sequence and hence if it is a monotonic sequence. (answer), Ex 11.2.4 Compute \(\sum_{n=0}^\infty {4\over (-3)^n}- {3\over 3^n}\). Ex 11.7.1 Compute \(\lim_{n\to\infty} |a_{n+1}/a_n|\) for the series \(\sum 1/n^2\). We will also determine a sequence is bounded below, bounded above and/or bounded. Determine whether the series converge or diverge. (answer), Ex 11.1.6 Determine whether \(\left\{{2^n\over n! /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 bmkraft7. /Name/F2 531.3 590.3 472.2 590.3 472.2 324.7 531.3 590.3 295.1 324.7 560.8 295.1 885.4 590.3 endobj /Name/F3 /Subtype/Type1 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). Ex 11.7.9 Prove theorem 11.7.3, the root test. Then click 'Next Question' to answer the next question. 24 0 obj Note that some sections will have more problems than others and some will have more or less of a variety of problems. 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 Research Methods Midterm. If you're seeing this message, it means we're having trouble loading external resources on our website. Which of the following is the 14th term of the sequence below? Quiz 1: 5 questions Practice what you've learned, and level up on the above skills. << (answer). When given a sum a[n], if the limit as n-->infinity does not exist or does not equal 0, the sum diverges. 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@ " 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. In the previous section, we determined the convergence or divergence of several series by . 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. >> 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 We will also give many of the basic facts and properties well need as we work with sequences. 777.8 444.4 444.4 444.4 611.1 777.8 777.8 777.8 777.8] /Filter /FlateDecode 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? Other sets by this creator. (5 points) Evaluate the integral: Z 1 1 1 x2 dx = SOLUTION: The function 1/x2 is undened at x = 0, so we we must evaluate the im- proper integral as a limit. Let the factor without dx equal u and the factor with dx equal dv. In order to use either test the terms of the infinite series must be positive. Ex 11.1.2 Use the squeeze theorem to show that \(\lim_{n\to\infty} {n!\over n^n}=0\). We will illustrate how we can find a series representation for indefinite integrals that cannot be evaluated by any other method. (answer), Ex 11.1.5 Determine whether \(\left\{{n+47\over\sqrt{n^2+3n}}\right\}_{n=1}^{\infty}\) converges or diverges. >> 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. 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, 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, Interval of convergence for derivative and integral, Integrals & derivatives of functions with known power series, 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. Complementary General calculus exercises can be found for other Textmaps and can be accessed here. Which equation below represents a geometric sequence? 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. 777.8 777.8] Proofs for both tests are also given. 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, \( \displaystyle \sum\limits_{n = 1}^\infty {\left( {n{2^n} - {3^{1 - n}}} \right)} \), \( \displaystyle \sum\limits_{n = 7}^\infty {\frac{{4 - n}}{{{n^2} + 1}}} \), \( \displaystyle \sum\limits_{n = 2}^\infty {\frac{{{{\left( { - 1} \right)}^{n - 3}}\left( {n + 2} \right)}}{{{5^{1 + 2n}}}}} \). 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 . << 252 0 obj <>stream Legal. Calculus II-Sequences and Series. (answer), Ex 11.1.4 Determine whether \(\left\{{n^2+1\over (n+1)^2}\right\}_{n=0}^{\infty}\) converges or diverges. Choose your answer to the question and click 'Continue' to see how you did. What if the interval is instead \([1,3/2]\)? Ex 11.7.2 Compute \(\lim_{n\to\infty} |a_{n+1}/a_n|\) for the series \(\sum 1/n\). Each term is the product of the two previous terms. /Name/F6 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_. Each term is the difference of the previous two terms. 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! 611.8 897.2 734 761.6 666.2 761.6 720.6 544 707.2 734 734 1006 734 734 598.4 272 Given that \( \displaystyle \sum\limits_{n = 0}^\infty {\frac{1}{{{n^3} + 1}}} = 1.6865\) determine the value of \( \displaystyle \sum\limits_{n = 2}^\infty {\frac{1}{{{n^3} + 1}}} \). Ratio Test In this section we will discuss using the Ratio Test to determine if an infinite series converges absolutely or diverges. Then click 'Next Question' to answer the next question. 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Some infinite series converge to a finite value. /FontDescriptor 11 0 R >> (answer), Ex 11.9.2 Find a power series representation for \(1/(1-x)^2\). xYKs6W(MCG:9iIO=(lkFRI$x$AMN/" J?~i~d cXf9o/r.&Lxy%/D-Yt+"LX]Sfp]Xl-aM_[6(*~mQbh*28AjZx0 =||. Sequences and Numerical series. Special Series In this section we will look at three series that either show up regularly or have some nice properties that we wish to discuss. >> 15 0 obj The following is a list of worksheets and other materials related to Math 129 at the UA. If it converges, compute the limit. 12 0 obj (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 \(\sum_{n=0}^N {1\over e^n} + 10^{-4}\). Divergence Test. 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. 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 /Subtype/Type1 About this unit. Khan Academy is a 501(c)(3) nonprofit organization. /Type/Font Psychological Research & Experimental Design, All Teacher Certification Test Prep Courses. Sequences and Series. Alternating Series Test - In this section we will discuss using the Alternating Series Test to determine if an infinite series converges or diverges. All rights reserved. stream 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 /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 << 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 . 5.3.3 Estimate the value of a series by finding bounds on its remainder term. The Alternating Series Test can be used only if the terms of the Absolute and conditional convergence. 1) \(\displaystyle \sum^_{n=1}a_n\) where \(a_n=\dfrac{2}{n . endobj 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 After each bounce, the ball reaches a height that is 2/3 of the height from which it previously fell. Here are a set of practice problems for the Series and Sequences chapter of the Calculus II notes. >> Donate or volunteer today! Accessibility StatementFor more information contact us atinfo@libretexts.org. 665 570.8 924.4 812.6 568.1 670.2 380.8 380.8 380.8 979.2 979.2 410.9 514 416.3 421.4 Convergence/Divergence of Series In this section we will discuss in greater detail the convergence and divergence of infinite series. /LastChar 127 /Type/Font We will also see how we can use the first few terms of a power series to approximate a function. Strip out the first 3 terms from the series n=1 2n n2 +1 n = 1 2 n n 2 + 1. stream Khan Academy is a 501(c)(3) nonprofit organization. %PDF-1.5 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 \(\sum_{n=1}^N {1\over n^4} + 0.005\). Here are a set of practice problems for the Series and Sequences chapter of the Calculus II notes. Donate or volunteer today! )^2\over n^n}\) (answer). /FontDescriptor 8 0 R 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! Images. 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. n a n converges if and only if the integral 1 f ( x) d x converges. stream 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. (answer), Ex 11.2.6 Compute \(\sum_{n=0}^\infty {4^{n+1}\over 5^n}\). 979.2 489.6 489.6 489.6] 4 avwo/MpLv) _C>5p*)i=^m7eE. Therefore the radius of convergence is R = , and the interval of convergence is ( - , ). If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. 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 /LastChar 127 /Length 1722 Given item A, which of the following would be the value of item B? 556.5 425.2 527.8 579.5 613.4 636.6 609.7 458.2 577.1 808.9 505 354.2 641.4 979.2 Level up on all the skills in this unit and collect up to 2000 Mastery points! /Subtype/Type1 /Type/Font My calculus 2 exam on sequence, infinite series & power seriesThe exam: https://bit.ly/36OHYcsAll the convergence tests: https://bit.ly/2IzqokhBest friend an. Then determine if the series converges or diverges. (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 \(\sum_{n=2}^N {1\over n(\ln n)^2} + 0.005\). 816 816 272 299.2 489.6 489.6 489.6 489.6 489.6 734 435.2 489.6 707.2 761.6 489.6 In other words, a series is the sum of a sequence. Ex 11.7.4 Compute \(\lim_{n\to\infty} |a_n|^{1/n}\) for the series \(\sum 1/n\). (You may want to use Sage or a similar aid.) (answer). Good luck! Alternating series test. Then we can say that the series diverges without having to do any extra work. Choose the equation below that represents the rule for the nth term of the following geometric sequence: 128, 64, 32, 16, 8, . %PDF-1.5 MATH 126 Medians and Such. 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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