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. 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. Indiana Core Assessments Mathematics: Test Prep & Study Guide. Maclaurin series of e, sin(x), and cos(x). (answer), Ex 11.2.2 Explain why \(\sum_{n=1}^\infty {5\over 2^{1/n}+14}\) diverges. 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. /Name/F5 After each bounce, the ball reaches a height that is 2/3 of the height from which it previously fell. Quiz 1: 5 questions Practice what you've learned, and level up on the above skills. Solution. The Alternating Series Test can be used only if the terms of the /Length 1722 Which is the finite sequence of four multiples of 9, starting with 9? Root Test In this section we will discuss using the Root Test to determine if an infinite series converges absolutely or diverges. Contact us by phone at (877)266-4919, or by mail at 100ViewStreet#202, MountainView, CA94041. /Subtype/Type1 Which equation below represents a geometric sequence? Ex 11.1.3 Determine whether \(\{\sqrt{n+47}-\sqrt{n}\}_{n=0}^{\infty}\) converges or diverges. endobj PDF Ap Calculus Ab Bc Kelley Copy - gny.salvationarmy.org 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 If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. Research Methods Midterm. x=S0 979.2 489.6 489.6 489.6] 2.(a). << (answer), Ex 11.2.6 Compute \(\sum_{n=0}^\infty {4^{n+1}\over 5^n}\). /BaseFont/CQGOFL+CMSY10 (You may want to use Sage or a similar aid.) endstream 207 0 obj <> endobj /FontDescriptor 14 0 R 668.3 724.7 666.7 666.7 666.7 666.7 666.7 611.1 611.1 444.4 444.4 444.4 444.4 500 888.9 888.9 888.9 888.9 888.9 888.9 888.9 666.7 875 875 875 875 611.1 611.1 833.3 |: The Ratio Test shows us that regardless of the choice of x, the series converges. /LastChar 127 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. Series Infinite geometric series: Series nth-term test: Series Integral test: Series Harmonic series and p-series: Series Comparison tests: . 24 0 obj Chapter 10 : Series and Sequences. If it converges, compute the limit. )^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 \(\ln 2\). }\) (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! Khan Academy is a 501(c)(3) nonprofit organization. We will also give many of the basic facts, properties and ways we can use to manipulate a series. Which of the following is the 14th term of the sequence below? In exercises 3 and 4, do not attempt to determine whether the endpoints are in the interval of convergence. . hbbd```b``~"A$" "Y`L6`RL,-`sA$w64= f[" RLMu;@jAl[`3H^Ne`?$4 << Images. endobj Math 1242: Calculus II - University of North Carolina at Charlotte 777.8 777.8] 21 terms. 5.3.2 Use the integral test to determine the convergence of a series. Complementary General calculus exercises can be found for other Textmaps and can be accessed here. A review of all series tests. /Filter /FlateDecode Then click 'Next Question' to answer the next question. 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 If you're seeing this message, it means we're having trouble loading external resources on our website. We will also give many of the basic facts and properties well need as we work with sequences. Solving My Calc 2 Exam#3 (Sequence, Infinite Series & Power Series) Series The Basics In this section we will formally define an infinite series. We will also give the Divergence Test for series in this section. Learning Objectives. 5.3.3 Estimate the value of a series by finding bounds on its remainder term. Each term is the difference of the previous two terms. When you have completed the free practice test, click 'View Results' to see your results. 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 What is the 83rd term of the sequence 91, 87, 83, 79, ( = a. Course summary; . stream Donate or volunteer today! stream 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 \(\sum_{n=1}^N {1\over n^4} + 0.005\). /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 If a geometric series begins with the following term, what would the next term be? /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 (answer), Ex 11.4.6 Approximate \(\sum_{n=1}^\infty (-1)^{n-1}{1\over n^4}\) to two decimal places. The steps are terms in the sequence. << Strip out the first 3 terms from the series n=1 2n n2 +1 n = 1 2 n n 2 + 1. 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. 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. 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 /Subtype/Type1 n = 1 n 2 + 2 n n 3 + 3 n . xTn0+,ITi](N@ fH2}W"UG'.% Z#>y{!9kJ+ Ex 11.1.2 Use the squeeze theorem to show that \(\lim_{n\to\infty} {n!\over n^n}=0\). /FirstChar 0 PDF Review Sheet for Calculus 2 Sequences and Series - Derrick Chung If it converges, compute the limit. Math 129 - Calculus II. /Filter /FlateDecode 272 816 544 489.6 544 516.8 380.8 386.2 380.8 544 516.8 707.2 516.8 516.8 435.2 489.6 500 388.9 388.9 277.8 500 500 611.1 500 277.8 833.3 750 833.3 416.7 666.7 666.7 777.8 666.7 1000 1000 1000 1000 1055.6 1055.6 1055.6 777.8 666.7 666.7 450 450 450 450 (answer), Ex 11.10.9 Use a combination of Maclaurin series and algebraic manipulation to find a series centered at zero for \( x\cos (x^2)\). Published by Wiley. 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. Which of the following sequences is NOT a geometric sequence? For each of the following series, determine which convergence test is the best to use and explain why. 6.5E: Exercises for Comparison Test - Mathematics LibreTexts << 1 2, 1 4, 1 8, Sequences of values of this type is the topic of this rst section. /FirstChar 0 531.3 531.3 531.3 295.1 295.1 295.1 826.4 501.7 501.7 826.4 795.8 752.1 767.4 811.1 Determine whether the following series converge or diverge. Worksheets. Learn for free about math, art, computer programming, economics, physics, chemistry, biology, medicine, finance, history, and more. endobj copyright 2003-2023 Study.com. /BaseFont/SFGTRF+CMSL12 Parametric equations, polar coordinates, and vector-valued functions Calculator-active practice: Parametric equations, polar coordinates, . Choose your answer to the question and click 'Continue' to see how you did. L7s[AQmT*Z;HK%H0yqt1r8 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 ]^e-V!2 F. 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 Comparison Test: This applies . Infinite series are sums of an infinite number of terms. /Filter /FlateDecode /Type/Font Strategies for Testing Series - University of Texas at Austin /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 We will examine Geometric Series, Telescoping Series, and Harmonic Series. /Type/Font >>
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