Quiz 2: 8 questions Practice what you've learned, and level up on the above skills. Calculus (single and multi-variable) Ordinary Differential equations (upto 2nd order linear with Laplace transforms, including Dirac Delta functions and Fourier Series. Part II. in calculus coursesincluding Calculus, Calculus II, Calculus III, AP Calculus and Precalculus. (answer). Therefore the radius of convergence is R = , and the interval of convergence is ( - , ). )^2\over n^n}\) (answer). &/ r
For each function, find the Maclaurin series or Taylor series centered at $a$, and the radius of convergence. >> Proofs for both tests are also given. /Type/Font (answer), Ex 11.2.5 Compute \(\sum_{n=0}^\infty {3\over 2^n}+ {4\over 5^n}\). n a n converges if and only if the integral 1 f ( x) d x converges. endstream
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<. /FirstChar 0 When you have completed the free practice test, click 'View Results' to see your results. Which rule represents the nth term in the sequence 9, 16, 23, 30? raVQ1CKD3`
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`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$&DN>+gt 1000 1000 1000 777.8 275 1000 666.7 666.7 888.9 888.9 0 0 555.6 555.6 666.7 500 722.2 665 570.8 924.4 812.6 568.1 670.2 380.8 380.8 380.8 979.2 979.2 410.9 514 416.3 421.4 stream 531.3 590.3 560.8 414.1 419.1 413.2 590.3 560.8 767.4 560.8 560.8 472.2 531.3 1062.5 x[[o6~cX/e`ElRm'1%J$%v)tb]1U2sRV}.l%s\Y UD+q}O+J Here are a set of practice problems for the Series and Sequences chapter of the Calculus II notes. Then determine if the series converges or diverges. Calculus II-Sequences and Series. Comparison tests. /Length 2492 These are homework exercises to accompany David Guichard's "General Calculus" Textmap. What is the sum of all the even integers from 2 to 250? 21 0 obj For problems 1 3 perform an index shift so that the series starts at \(n = 3\). (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 \(\sum_{n=1}^N {\ln n\over n^2} + 0.005\). 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 . Harmonic series and p-series. /FontDescriptor 20 0 R ]^e-V!2
F. Chapter 10 : Series and Sequences. Infinite series are sums of an infinite number of terms. 979.2 489.6 489.6 489.6] A proof of the Ratio Test is also given. << >> When given a sum a[n], if the limit as n-->infinity does not exist or does not equal 0, the sum diverges. L7s[AQmT*Z;HK%H0yqt1r8 The practice tests are composed 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). Sequences and Series. Applications of Series In this section we will take a quick look at a couple of applications of series. Level up on all the skills in this unit and collect up to 2000 Mastery points! /FirstChar 0 What if the interval is instead \([1,3/2]\)? When you have completed the free practice test, click 'View Results' to see your results. Psychological Research & Experimental Design, All Teacher Certification Test Prep Courses. Don't all infinite series grow to infinity? 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 . To log in and use all the features of Khan Academy, please enable JavaScript in your browser. /Subtype/Type1 %PDF-1.2 Ex 11.7.2 Compute \(\lim_{n\to\infty} |a_{n+1}/a_n|\) for the series \(\sum 1/n\). Published by Wiley. Remark. /LastChar 127 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? 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. 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 xYKs6W(MCG:9iIO=(lkFRI$x$AMN/"
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}\) (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! Math C185: Calculus II (Tran) 6: Sequences and Series 6.5: Comparison Tests 6.5E: Exercises for Comparison Test Expand/collapse global location 6.5E: Exercises for Comparison Test . 666.7 1000 1000 1000 1000 1055.6 1055.6 1055.6 777.8 666.7 666.7 450 450 450 450 /Name/F2 If you . /Type/Font Which of the following is the 14th term of the sequence below? Root Test In this section we will discuss using the Root Test to determine if an infinite series converges absolutely or diverges. 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. Which of the following sequences is NOT a geometric sequence? Which is the finite sequence of four multiples of 9, starting with 9? Which of the following is the 14th term of the sequence below? (answer), Ex 11.2.2 Explain why \(\sum_{n=1}^\infty {5\over 2^{1/n}+14}\) diverges. What is the radius of convergence? 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. Which of the following sequences follows this formula. stream 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 . Determine whether the following series converge or diverge. 1. 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. (1 point) Is the integral Z 1 1 1 x2 dx an improper integral? The Alternating Series Test can be used only if the terms of the series alternate in sign. OR sequences are lists of numbers, where the numbers may or may not be determined by a pattern. All rights reserved. << 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 Parametric equations, polar coordinates, and vector-valued functions Calculator-active practice: Parametric equations, polar coordinates, . n = 1 n2 + 2n n3 + 3n2 + 1. >> /FirstChar 0 500 388.9 388.9 277.8 500 500 611.1 500 277.8 833.3 750 833.3 416.7 666.7 666.7 777.8 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! 1277.8 555.6 1000 1444.4 555.6 1000 1444.4 472.2 472.2 527.8 527.8 527.8 527.8 666.7 /Length 200 816 816 272 299.2 489.6 489.6 489.6 489.6 489.6 792.7 435.2 489.6 707.2 761.6 489.6 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 . We will determine if a sequence in an increasing sequence or a decreasing sequence and hence if it is a monotonic sequence. /Widths[606.7 816 748.3 679.6 728.7 811.3 765.8 571.2 652.8 598 757.6 622.8 552.8 805.6 805.6 1277.8 1277.8 811.1 811.1 875 875 666.7 666.7 666.7 666.7 666.7 666.7 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. /Subtype/Type1 /FontDescriptor 17 0 R 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. 4 avwo/MpLv)
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It turns out the answer is no. /FontDescriptor 23 0 R 21 terms. Indiana Core Assessments Mathematics: Test Prep & Study Guide. 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 xu? ~k"xPeEV4Vcwww \ a:5d*%30EU9>,e92UU3Voj/$f
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We also discuss differentiation and integration of power series. Khan Academy is a 501(c)(3) nonprofit organization. Learning Objectives. 17 0 obj To log in and use all the features of Khan Academy, please enable JavaScript in your browser. 252 0 obj
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(You may want to use Sage or a similar aid.) 883.8 992.6 761.6 272 272 489.6 816 489.6 816 761.6 272 380.8 380.8 489.6 761.6 272 979.2 489.6 489.6 489.6] (b) /Length 1247 /LastChar 127 endstream 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 11.1.5 Determine whether \(\left\{{n+47\over\sqrt{n^2+3n}}\right\}_{n=1}^{\infty}\) converges or diverges. 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. Determine whether the series converge or diverge. /Widths[458.3 458.3 416.7 416.7 472.2 472.2 472.2 472.2 583.3 583.3 472.2 472.2 333.3 826.4 531.3 958.7 1076.8 826.4 295.1 295.1 531.3 885.4 531.3 885.4 826.4 295.1 413.2 Premium members get access to this practice exam along with our entire library of lessons taught by subject matter experts. For problems 1 - 3 perform an index shift so that the series starts at n = 3 n = 3. (answer), Ex 11.2.3 Explain why \(\sum_{n=1}^\infty {3\over n}\) diverges. The steps are terms in the sequence. Differentiate u to find du, and integrate dv to find v. Use the formula: Evaluate the right side of this equation to solve the integral. Learn for free about math, art, computer programming, economics, physics, chemistry, biology, medicine, finance, history, and more. /FontDescriptor 11 0 R % /BaseFont/PSJLQR+CMEX10 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. 777.8 444.4 444.4 444.4 611.1 777.8 777.8 777.8 777.8] /Type/Font (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 \(\sum_{n=0}^N {1\over e^n} + 10^{-4}\). /Subtype/Type1 /Filter /FlateDecode The chapter headings refer to Calculus, Sixth Edition by Hughes-Hallett et al. /Filter[/FlateDecode] We also derive some well known formulas for Taylor series of \({\bf e}^{x}\) , \(\cos(x)\) and \(\sin(x)\) around \(x=0\). /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 Legal. stream bmkraft7. )Ltgx?^eaT'&+n+hN4*D^UR!8UY@>LqS%@Cp/-12##DR}miBw6"ja+WjU${IH$5j!j-I1 |: The Ratio Test shows us that regardless of the choice of x, the series converges. 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 Complementary General calculus exercises can be found for other Textmaps and can be accessed here. A brick wall has 60 bricks in the first row, but each row has 3 fewer bricks than the previous one. Let the factor without dx equal u and the factor with dx equal dv. A proof of the Root Test is also given. ZrNRG{I~(iw%0W5b)8*^
yyCCy~Cg{C&BPsTxp%p Convergence/Divergence of Series In this section we will discuss in greater detail the convergence and divergence of infinite series. Series Infinite geometric series: Series nth-term test: Series Integral test: Series Harmonic series and p-series: Series Comparison tests: . The book contains eight practice tests five practice tests for Calculus AB and three practice tests for Calculus BC. 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 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. Then click 'Next Question' to answer the . /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 1 2, 1 4, 1 8, Sequences of values of this type is the topic of this rst section. (answer), Ex 11.2.4 Compute \(\sum_{n=0}^\infty {4\over (-3)^n}- {3\over 3^n}\). 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, \( \displaystyle \sum\limits_{n = 1}^\infty {\left( {n{2^n} - {3^{1 - n}}} \right)} \), \( \displaystyle \sum\limits_{n = 7}^\infty {\frac{{4 - n}}{{{n^2} + 1}}} \), \( \displaystyle \sum\limits_{n = 2}^\infty {\frac{{{{\left( { - 1} \right)}^{n - 3}}\left( {n + 2} \right)}}{{{5^{1 + 2n}}}}} \). Which of the following sequences follows this formula? Here are a set of practice problems for the Series and Sequences chapter of the Calculus II notes. 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. /BaseFont/SFGTRF+CMSL12 Which of the sequences below has the recursive rule where each number is the previous number times 2? !A1axw)}p]WgxmkFftu (answer). (answer), Ex 11.2.6 Compute \(\sum_{n=0}^\infty {4^{n+1}\over 5^n}\). 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 Note that some sections will have more problems than others and some will have more or less of a variety of problems. 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 Our mission is to provide a free, world-class education to anyone, anywhere. endstream 611.8 897.2 734 761.6 666.2 761.6 720.6 544 707.2 734 734 1006 734 734 598.4 272 endobj If the series is an alternating series, determine whether it converges absolutely, converges conditionally, or diverges. . In the previous section, we determined the convergence or divergence of several series by . endobj (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. Free Practice Test Instructions: Choose your answer to the question and click 'Continue' to see how you did. 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 << >> If it con-verges, nd the limit. Choose your answer to the question and click 'Continue' to see how you did. /LastChar 127 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. 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@ "
/Name/F4 Here are a set of practice problems for the Series and Sequences chapter of the Calculus II notes. The following is a list of worksheets and other materials related to Math 129 at the UA. Khan Academy is a 501(c)(3) nonprofit organization. >> Psychological Research & Experimental Design, All Teacher Certification Test Prep Courses. Don't all infinite series grow to infinity? Consider the series n a n. Divergence Test: If lim n a n 0, then n a n diverges. Ex 11.8.1 \(\sum_{n=0}^\infty n x^n\) (answer), Ex 11.8.2 \(\sum_{n=0}^\infty {x^n\over n! /LastChar 127 We will also see how we can use the first few terms of a power series to approximate a function. Final: all from 02/05 and 03/11 exams (except work, separation of variables, and probability) plus sequences, series, convergence tests, power series, Taylor series. Images. Calculus 2. If youd like to view the solutions on the web go to the problem set web page, click the solution link for any problem and it will take you to the solution to that problem. Determine whether each series converges absolutely, converges conditionally, or diverges. Some infinite series converge to a finite value. YesNo 2.(b). Worksheets. 2 6 points 2. Good luck! Sequences and Numerical series. Each term is the sum of the previous two terms. endobj MATH 126 Medians and Such. %|S#?\A@D-oS)lW=??nn}y]Tb!!o_=;]ha,J[. %PDF-1.5 Maclaurin series of e, sin(x), and cos(x). }\right\}_{n=0}^{\infty}\) converges or diverges. 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. 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 Ex 11.7.3 Compute \(\lim_{n\to\infty} |a_n|^{1/n}\) for the series \(\sum 1/n^2\). << ,vEmO8/OuNVRaLPqB.*l. If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. Choose your answer to the question and click 'Continue' to see how you did. 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. Sequences can be thought of as functions whose domain is the set of integers. 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 (answer). All other trademarks and copyrights are the property of their respective owners. %PDF-1.5 If L = 1, then the test is inconclusive. Then click 'Next Question' to answer the next question. Ex 11.1.3 Determine whether {n + 47 n} . endstream
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About this unit. 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 11.1.6 Determine whether \(\left\{{2^n\over n! 12 0 obj >> 1000 500 500 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 UcTIjeB#vog-TM'FaTzG(:k-BNQmbj}'?^h<=XgS/]o4Ilv%Jm /LastChar 127 sCA%HGEH[ Ah)lzv<7'9&9X}xbgY[ xI9i,c_%tz5RUam\\6(ke9}Yv`B7yYdWrJ{KZVUYMwlbN_>[wle\seUy24P,PyX[+l\c $w^rvo]cYc@bAlfi6);;wOF&G_. /Filter /FlateDecode /Filter /FlateDecode )^2\over n^n}(x-2)^n\) (answer), Ex 11.8.6 \(\sum_{n=1}^\infty {(x+5)^n\over n(n+1)}\) (answer), Ex 11.9.1 Find a series representation for \(\ln 2\). If you're seeing this message, it means we're having trouble loading external resources on our website. Course summary; . xWKoFWlojCpP NDED$(lq"g|3g6X_&F1BXIM5d gOwaN9c,r|9 Other sets by this creator. 1) \(\displaystyle \sum^_{n=1}a_n\) where \(a_n=\dfrac{2}{n . /BaseFont/CQGOFL+CMSY10 Most sections should have a range of difficulty levels in the problems although this will vary from section to section. /Name/F6 If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. 8 0 obj (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 \( 10^{-3}\)? Integral Test: If a n = f ( n), where f ( x) is a non-negative non-increasing function, then. Infinite series are sums of an infinite number of terms. /Type/Font 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 (answer), Ex 11.1.4 Determine whether \(\left\{{n^2+1\over (n+1)^2}\right\}_{n=0}^{\infty}\) converges or diverges. 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 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. (answer), Ex 11.10.10 Use a combination of Maclaurin series and algebraic manipulation to find a series centered at zero for \( xe^{-x}\). endobj 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 Math 106 (Calculus II): old exams. 62 0 obj In order to use either test the terms of the infinite series must be positive. Comparison Test: This applies . A ball is dropped from an unknown height (h) and it repeatedly bounces on the floor. AP is a registered trademark of the College Board, which has not reviewed this resource. /BaseFont/VMQJJE+CMR8 The Integral Test can be used on a infinite series provided the terms of the series are positive and decreasing. /Type/Font We will also give many of the basic facts, properties and ways we can use to manipulate a series. 1 2 + 1 4 + 1 8 + = n=1 1 2n = 1 We will need to be careful, but it turns out that we can . Strip out the first 3 terms from the series n=1 2n n2 +1 n = 1 2 n n 2 + 1. If a geometric series begins with the following term, what would the next term be? Khan Academy is a nonprofit with the mission of providing a free, world-class education for anyone, anywhere. 26 0 obj (answer), Ex 11.9.4 Find a power series representation for \( 1/(1-x)^3\). Learn how this is possible, how we can tell whether a series converges, and how we can explore convergence in Taylor and Maclaurin series. 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 At this time, I do not offer pdfs for solutions to individual problems. A proof of the Alternating Series Test is also given. Remark.
Martha_Austin Teacher. (answer), Ex 11.2.8 Compute \(\sum_{n=1}^\infty \left({3\over 5}\right)^n\). Ratio Test In this section we will discuss using the Ratio Test to determine if an infinite series converges absolutely or diverges. 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. 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 (answer). endobj %PDF-1.5
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My calculus 2 exam on sequence, infinite series & power seriesThe exam: https://bit.ly/36OHYcsAll the convergence tests: https://bit.ly/2IzqokhBest friend an. Ex 11.7.1 Compute \(\lim_{n\to\infty} |a_{n+1}/a_n|\) for the series \(\sum 1/n^2\). 5.3.2 Use the integral test to determine the convergence of a series. Integral Test In this section we will discuss using the Integral Test to determine if an infinite 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 \(\sum_{n=2}^N {1\over n(\ln n)^2} + 0.005\). 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. Then click 'Next Question' to answer the next question. (answer), Ex 11.2.7 Compute \(\sum_{n=0}^\infty {3^{n+1}\over 7^{n+1}}\). { "11.01:_Prelude_to_Sequences_and_Series" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.
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