/Length 569 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. Chapter 10 : Series and Sequences. /Subtype/Type1 stream 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. If a geometric series begins with the following term, what would the next term be? Alternating series test. Which of the following sequences follows this formula? /Filter[/FlateDecode] 11.E: Sequences and Series (Exercises) These are homework exercises to accompany David Guichard's "General Calculus" Textmap. (answer), Ex 11.2.1 Explain why \(\sum_{n=1}^\infty {n^2\over 2n^2+1}\) diverges. << >> (answer), Ex 11.2.6 Compute \(\sum_{n=0}^\infty {4^{n+1}\over 5^n}\). At this time, I do not offer pdf's for solutions to individual problems. Alternating Series Test - In this section we will discuss using the Alternating Series Test to determine if an infinite series converges or diverges. 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 )^2\over n^n}\) (answer). 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 Ex 11.7.3 Compute \(\lim_{n\to\infty} |a_n|^{1/n}\) for the series \(\sum 1/n^2\). MATH 126 Medians and Such. Calculus II-Sequences and Series. We also discuss differentiation and integration of power series. (answer), Ex 11.2.4 Compute \(\sum_{n=0}^\infty {4\over (-3)^n}- {3\over 3^n}\). (answer), Ex 11.1.4 Determine whether \(\left\{{n^2+1\over (n+1)^2}\right\}_{n=0}^{\infty}\) converges or diverges. We will also determine a sequence is bounded below, bounded above and/or bounded. A Lot of Series Test Practice Problems - YouTube (1 point) Is the integral Z 1 1 1 x2 dx an improper integral? >> We will illustrate how partial sums are used to determine if an infinite series converges or diverges. Solution. Ex 11.1.2 Use the squeeze theorem to show that limn n! bmkraft7. PDF M 172 - Calculus II - Chapter 10 Sequences and 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. 8 0 obj You may also use any of these materials for practice. 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. Choose your answer to the question and click 'Continue' to see how you did. If it converges, compute the limit. We will also give many of the basic facts, properties and ways we can use to manipulate a series. Which one of these sequences is a finite sequence? Calculus II - Series - The Basics (Practice Problems) - Lamar University 68 0 obj AP Calculus AB and BC: Chapter 9 -Infinite Sequences and Series : 9.4 Sequences review (practice) | Series | Khan Academy Divergence Test. (answer), Ex 11.10.9 Use a combination of Maclaurin series and algebraic manipulation to find a series centered at zero for \( x\cos (x^2)\). Infinite sequences and series | AP/College Calculus BC - Khan Academy Determine whether each series converges or diverges. /BaseFont/PSJLQR+CMEX10 PDF FINAL EXAM CALCULUS 2 - Department of Mathematics Find the radius and interval of convergence for each series. 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 . 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 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. 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. 666.7 1000 1000 1000 1000 1055.6 1055.6 1055.6 777.8 666.7 666.7 450 450 450 450 YesNo 2.(b). The sum of the steps forms an innite series, the topic of Section 10.2 and the rest of Chapter 10. Donate or volunteer today! /Name/F5 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 (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 \(\sum_{n=1}^N {\ln n\over n^2} + 0.005\). endobj The following is a list of worksheets and other materials related to Math 129 at the UA. 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 All rights reserved. A proof of the Ratio Test is also given. 31 terms. Some infinite series converge to a finite value. 508.8 453.8 482.6 468.9 563.7 334 405.1 509.3 291.7 856.5 584.5 470.7 491.4 434.1 207 0 obj
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endstream SAT Practice Questions- All Maths; SAT Practice Test Questions- Reading , Writing and Language; KS 1-2 Math, Science and SAT . >> In order to use either test the terms of the infinite series must be positive. The chapter headings refer to Calculus, Sixth Edition by Hughes-Hallett et al. All other trademarks and copyrights are the property of their respective owners. Published by Wiley. hbbd```b``~"A$"
"Y`L6`RL,-`sA$w64= f[" RLMu;@jAl[`3H^Ne`?$4 /LastChar 127 5.3.3 Estimate the value of a series by finding bounds on its remainder term. 9 0 obj 18 0 obj Math 106 (Calculus II): old exams | Mathematics | Bates College Strip out the first 3 terms from the series n=1 2n n2 +1 n = 1 2 n n 2 + 1. Calculus 2 | Math | Khan Academy /Subtype/Type1 Don't all infinite series grow to infinity? 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 . To use the Geometric Series formula, the function must be able to be put into a specific form, which is often impossible. 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. 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. 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, \( \displaystyle \sum\limits_{n = 1}^\infty {\left( {n{2^n} - {3^{1 - n}}} \right)} \), \( \displaystyle \sum\limits_{n = 7}^\infty {\frac{{4 - n}}{{{n^2} + 1}}} \), \( \displaystyle \sum\limits_{n = 2}^\infty {\frac{{{{\left( { - 1} \right)}^{n - 3}}\left( {n + 2} \right)}}{{{5^{1 + 2n}}}}} \). %%EOF
/LastChar 127 Khan Academy is a 501(c)(3) nonprofit organization. What is the 83rd term of the sequence 91, 87, 83, 79, ( = a. Most sections should have a range of difficulty levels in the problems although this will vary from section to section. These are homework exercises to accompany David Guichard's "General Calculus" Textmap. 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. (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. endobj 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. UcTIjeB#vog-TM'FaTzG(:k-BNQmbj}'?^h<=XgS/]o4Ilv%Jm 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. endobj PDF Calculus II Series - Things to Consider - California State University % 45 0 obj /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 \ _* %l~G"tytO(J*l+X@ uE: m/ ~&Q24Nss(7F!ky=4 Mijo8t;v 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. Comparison tests. Learn how this is possible, how we can tell whether a series converges, and how we can explore convergence in . 611.8 897.2 734 761.6 666.2 761.6 720.6 544 707.2 734 734 1006 734 734 598.4 272 Sequences and Numerical series. 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 Proofs for both tests are also given. /Length 465 Complementary General calculus exercises can be found for other Textmaps and can be accessed here. /Subtype/Type1 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. When you have completed the free practice test, click 'View Results' to see your results. ]^e-V!2
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