We will also see how we can use the first few terms of a power series to approximate a function. 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. xYKs6W(MCG:9iIO=(lkFRI$x$AMN/"
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cXf9o/r.&Lxy%/D-Yt+"LX]Sfp]Xl-aM_[6(*~mQbh*28AjZx0 =||. Determine whether the following series converge or diverge. To use integration by parts in Calculus, follow these steps: Decompose the entire integral (including dx) into two factors. /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 << /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 Khan Academy is a 501(c)(3) nonprofit organization. 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 It turns out the answer is no. (answer). The chapter headings refer to Calculus, Sixth Edition by Hughes-Hallett et al. 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. (b) << Which of the following is the 14th term of the sequence below? /Subtype/Type1 >> Alternating series test. Given that \( \displaystyle \sum\limits_{n = 0}^\infty {\frac{1}{{{n^3} + 1}}} = 1.6865\) determine the value of \( \displaystyle \sum\limits_{n = 2}^\infty {\frac{1}{{{n^3} + 1}}} \). Free Practice Test Instructions: Choose your answer to the question and click 'Continue' to see how you did. To log in and use all the features of Khan Academy, please enable JavaScript in your browser. The steps are terms in the sequence. 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 . (answer), Ex 11.2.3 Explain why \(\sum_{n=1}^\infty {3\over n}\) diverges. endstream
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<. Which of the following sequences is NOT a geometric sequence? /Subtype/Type1 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. 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. << (answer), Ex 11.1.6 Determine whether \(\left\{{2^n\over n!
L7s[AQmT*Z;HK%H0yqt1r8 /Length 1247 Use the Comparison Test to determine whether each series in exercises 1 - 13 converges or diverges. For problems 1 3 perform an index shift so that the series starts at \(n = 3\). 68 0 obj /Type/Font 2 6 points 2. Premium members get access to this practice exam along with our entire library of lessons taught by subject matter experts. The following is a list of worksheets and other materials related to Math 129 at the UA. Ex 11.11.4 Show that \(\cos x\) is equal to its Taylor series for all \(x\) by showing that the limit of the error term is zero as N approaches infinity. Ratio test. (answer), Ex 11.2.6 Compute \(\sum_{n=0}^\infty {4^{n+1}\over 5^n}\). )^2\over n^n}\) (answer). Accessibility StatementFor more information contact us [email protected]. }\) (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! 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 /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 In other words, a series is the sum of a sequence. /FontDescriptor 23 0 R 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 Determine whether the sequence converges or diverges. %PDF-1.5 The sum of the steps forms an innite series, the topic of Section 10.2 and the rest of Chapter 10. Ex 11.10.8 Find the first four terms of the Maclaurin series for \(\tan x\) (up to and including the \( x^3\) term). % %|S#?\A@D-oS)lW=??nn}y]Tb!!o_=;]ha,J[. . Published by Wiley. Psychological Research & Experimental Design, All Teacher Certification Test Prep Courses. When you have completed the free practice test, click 'View Results' to see your results. 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@ "
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 Series The Basics In this section we will formally define an infinite series. %PDF-1.5 1000 1000 777.8 777.8 1000 1000 500 500 1000 1000 1000 777.8 1000 1000 611.1 611.1 stream (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 \(\sum_{n=2}^N {1\over n(\ln n)^2} + 0.005\). Series are sums of multiple terms. Determine whether the series converge or diverge. /FontDescriptor 11 0 R /LastChar 127 Alternating Series Test - In this section we will discuss using the Alternating Series Test to determine if an infinite series converges or diverges. 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 &/ r
If L = 1, then the test is inconclusive. Choose your answer to the question and click 'Continue' to see how you did. 1. 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 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 . Maclaurin series of e, sin(x), and cos(x). (answer), Ex 11.4.6 Approximate \(\sum_{n=1}^\infty (-1)^{n-1}{1\over n^4}\) to two decimal places. /BaseFont/VMQJJE+CMR8 Ex 11.7.9 Prove theorem 11.7.3, the root test. Ex 11.7.3 Compute \(\lim_{n\to\infty} |a_n|^{1/n}\) for the series \(\sum 1/n^2\). )Ltgx?^eaT'&+n+hN4*D^UR!8UY@>LqS%@Cp/-12##DR}miBw6"ja+WjU${IH$5j!j-I1 Choose your answer to the question and click 'Continue' to see how you did. 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 We will illustrate how partial sums are used to determine if an infinite series converges or diverges. 62 0 obj If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. Which of the following sequences follows this formula? Applications of Series In this section we will take a quick look at a couple of applications of series. 531.3 531.3 531.3] Here are a set of practice problems for the Series and Sequences chapter of the Calculus II notes. The book contains eight practice tests five practice tests for Calculus AB and three practice tests for Calculus BC. 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 Comparison Test: This applies . 531.3 531.3 531.3 295.1 295.1 295.1 826.4 501.7 501.7 826.4 795.8 752.1 767.4 811.1 777.8 777.8] The numbers used come from a sequence. /Filter /FlateDecode |: The Ratio Test shows us that regardless of the choice of x, the series converges. (answer), Ex 11.1.4 Determine whether \(\left\{{n^2+1\over (n+1)^2}\right\}_{n=0}^{\infty}\) converges or diverges. 8 0 obj Remark. 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
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_. Series Infinite geometric series: Series nth-term test: Series Integral test: Series Harmonic series and p-series: Series Comparison tests: . 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 /FirstChar 0 %PDF-1.2 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? /LastChar 127 Premium members get access to this practice exam along with our entire library of lessons taught by subject matter experts. Question 5 5. For problems 1 - 3 perform an index shift so that the series starts at n = 3 n = 3. The Alternating Series Test can be used only if the terms of the /LastChar 127 /Filter /FlateDecode Remark. /Filter /FlateDecode 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. /FirstChar 0 My calculus 2 exam on sequence, infinite series & power seriesThe exam: https://bit.ly/36OHYcsAll the convergence tests: https://bit.ly/2IzqokhBest friend an. 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 In the previous section, we determined the convergence or divergence of several series by . stream Learn for free about math, art, computer programming, economics, physics, chemistry, biology, medicine, finance, history, and more. Ratio Test In this section we will discuss using the Ratio Test to determine if an infinite series converges absolutely or diverges. When given a sum a[n], if the limit as n-->infinity does not exist or does not equal 0, the sum diverges. Indiana Core Assessments Mathematics: Test Prep & Study Guide. 611.8 897.2 734 761.6 666.2 761.6 720.6 544 707.2 734 734 1006 734 734 598.4 272 Ex 11.9.5 Find a power series representation for \(\int\ln(1-x)\,dx\). Good luck! Then we can say that the series diverges without having to do any extra work. %PDF-1.5
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Which one of these sequences is a finite sequence? /Length 569 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. 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. Solution. 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 . 15 0 obj << Calculus II-Sequences and Series. Taylor Series In this section we will discuss how to find the Taylor/Maclaurin Series for a function. 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 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 \(\sum_{n=1}^N {1\over n^4} + 0.005\). (You may want to use Sage or a similar aid.) Level up on all the skills in this unit and collect up to 2000 Mastery points! /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 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 /BaseFont/BPHBTR+CMMI12 (answer), Ex 11.2.8 Compute \(\sum_{n=1}^\infty \left({3\over 5}\right)^n\). >> Calculus (single and multi-variable) Ordinary Differential equations (upto 2nd order linear with Laplace transforms, including Dirac Delta functions and Fourier Series. We will examine Geometric Series, Telescoping Series, and Harmonic Series. Ex 11.7.4 Compute \(\lim_{n\to\infty} |a_n|^{1/n}\) for the series \(\sum 1/n\). After each bounce, the ball reaches a height that is 2/3 of the height from which it previously fell. What is the radius of convergence? What if the interval is instead \([1,3/2]\)? /LastChar 127 11.E: Sequences and Series (Exercises) These are homework exercises to accompany David Guichard's "General Calculus" Textmap. 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! Then click 'Next Question' to answer the . /Name/F2 We also derive some well known formulas for Taylor series of \({\bf e}^{x}\) , \(\cos(x)\) and \(\sin(x)\) around \(x=0\). Given item A, which of the following would be the value of item B? Khan Academy is a 501(c)(3) nonprofit organization. OR sequences are lists of numbers, where the numbers may or may not be determined by a pattern. copyright 2003-2023 Study.com. A proof of the Integral Test is also given. Choose your answer to the question and click 'Continue' to see how you did. Quiz 2: 8 questions Practice what you've learned, and level up on the above skills. Integral Test: If a n = f ( n), where f ( x) is a non-negative non-increasing function, then. Contact us by phone at (877)266-4919, or by mail at 100ViewStreet#202, MountainView, CA94041. n = 1 n 2 + 2 n n 3 + 3 n . 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. 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] /Type/Font >> Ex 11.7.2 Compute \(\lim_{n\to\infty} |a_{n+1}/a_n|\) for the series \(\sum 1/n\). If you're seeing this message, it means we're having trouble loading external resources on our website. 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 \ _* %l~G"tytO(J*l+X@ uE: m/ ~&Q24Nss(7F!ky=4 Mijo8t;v 252 0 obj
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/Subtype/Type1 590.3 767.4 795.8 795.8 1091 795.8 795.8 649.3 295.1 531.3 295.1 531.3 295.1 295.1 Worksheets. If it converges, compute the limit. Sequences & Series in Calculus Chapter Exam. 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 AP is a registered trademark of the College Board, which has not reviewed this resource. >> Strip out the first 3 terms from the series n=1 2n n2 +1 n = 1 2 n n 2 + 1. 979.2 489.6 489.6 489.6] 70 terms. 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). 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. Derivatives, Integrals, Sequences & Series, and Vector Valued Functions. >> endobj Determine whether the series is convergent or divergent. 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 . 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. Learn how this is possible, how we can tell whether a series converges, and how we can explore convergence in Taylor and Maclaurin series. Determine whether each series converges absolutely, converges conditionally, or diverges. Most sections should have a range of difficulty levels in the problems although this will vary from section to section. In exercises 3 and 4, do not attempt to determine whether the endpoints are in the interval of convergence. >> At this time, I do not offer pdfs for solutions to individual problems. (answer), Ex 11.2.5 Compute \(\sum_{n=0}^\infty {3\over 2^n}+ {4\over 5^n}\). 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 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, \( \displaystyle \sum\limits_{n = 1}^\infty {\left( {n{2^n} - {3^{1 - n}}} \right)} \), \( \displaystyle \sum\limits_{n = 7}^\infty {\frac{{4 - n}}{{{n^2} + 1}}} \), \( \displaystyle \sum\limits_{n = 2}^\infty {\frac{{{{\left( { - 1} \right)}^{n - 3}}\left( {n + 2} \right)}}{{{5^{1 + 2n}}}}} \). /FontDescriptor 8 0 R 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 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 MULTIPLE CHOICE: Circle the best answer. xWKoFWlojCpP NDED$(lq"g|3g6X_&F1BXIM5d gOwaN9c,r|9 /Length 200 hbbd```b``~"A$"
"Y`L6`RL,-`sA$w64= f[" RLMu;@jAl[`3H^Ne`?$4 /FirstChar 0 Math 129 - Calculus II. Research Methods Midterm. 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. Contact us by phone at (877)266-4919, or by mail at 100ViewStreet#202, MountainView, CA94041. !A1axw)}p]WgxmkFftu 777.8 444.4 444.4 444.4 611.1 777.8 777.8 777.8 777.8] Each term is the product of the two previous terms. Which is the finite sequence of four multiples of 9, starting with 9? When you have completed the free practice test, click 'View Results' to see your results. << The practice tests are composed /Length 2492 /BaseFont/PSJLQR+CMEX10 Choose your answer to the question and click 'Continue' to see how you did. ]^e-V!2
F. (answer). stream endobj MATH 126 Medians and Such. Convergence/Divergence of Series In this section we will discuss in greater detail the convergence and divergence of infinite series. We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. % Complementary General calculus exercises can be found for other Textmaps and can be accessed here. Ex 11.1.3 Determine whether \(\{\sqrt{n+47}-\sqrt{n}\}_{n=0}^{\infty}\) converges or diverges. Good luck! Note as well that there really isnt one set of guidelines that will always work and so you always need to be flexible in following this set of guidelines. YesNo 2.(b). /LastChar 127 Alternating Series Test In this section we will discuss using the Alternating Series Test to determine if an infinite series converges or diverges. endobj 17 0 obj Then click 'Next Question' to answer the next question. /Type/Font Proofs for both tests are also given. Ex 11.5.1 \(\sum_{n=1}^\infty {1\over 2n^2+3n+5} \) (answer), Ex 11.5.2 \(\sum_{n=2}^\infty {1\over 2n^2+3n-5} \) (answer), Ex 11.5.3 \(\sum_{n=1}^\infty {1\over 2n^2-3n-5} \) (answer), Ex 11.5.4 \(\sum_{n=1}^\infty {3n+4\over 2n^2+3n+5} \) (answer), Ex 11.5.5 \(\sum_{n=1}^\infty {3n^2+4\over 2n^2+3n+5} \) (answer), Ex 11.5.6 \(\sum_{n=1}^\infty {\ln n\over n}\) (answer), Ex 11.5.7 \(\sum_{n=1}^\infty {\ln n\over n^3}\) (answer), Ex 11.5.8 \(\sum_{n=2}^\infty {1\over \ln n}\) (answer), Ex 11.5.9 \(\sum_{n=1}^\infty {3^n\over 2^n+5^n}\) (answer), Ex 11.5.10 \(\sum_{n=1}^\infty {3^n\over 2^n+3^n}\) (answer). A proof of the Root Test is also given. 1 2, 1 4, 1 8, Sequences of values of this type is the topic of this rst section. 21 0 obj (answer). << Parametric equations, polar coordinates, and vector-valued functions Calculator-active practice: Parametric equations, polar coordinates, . 9 0 obj 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. 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 . /BaseFont/CQGOFL+CMSY10 A proof of the Alternating Series Test is also given. /Name/F3 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. /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 Donate or volunteer today! 26 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 \( 10^{-3}\)? (1 point) Is the integral Z 1 1 1 x2 dx an improper integral? Images. >> For each function, find the Maclaurin series or Taylor series centered at $a$, and the radius of convergence. /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 \(\sum_{n=0}^N {1\over e^n} + 10^{-4}\). 590.3 885.4 885.4 295.1 324.7 531.3 531.3 531.3 531.3 531.3 795.8 472.2 531.3 767.4 500 388.9 388.9 277.8 500 500 611.1 500 277.8 833.3 750 833.3 416.7 666.7 666.7 777.8 << /LastChar 127 /Name/F4 A review of all series tests. 4 avwo/MpLv)
_C>5p*)i=^m7eE. Integral test. (a) $\sum_{n=1}^{\infty} \frac{(-1)^n}{\sqrt{n}}$ (b) $\sum_{n=1}^{\infty}(-1)^n \frac{n}{2 n-1}$ In order to use either test the terms of the infinite series must be positive. When you have completed the free practice test, click 'View Results' to see your results. /BaseFont/SFGTRF+CMSL12 { "11.01:_Prelude_to_Sequences_and_Series" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.
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