/FontDescriptor 17 0 R If you . 24 0 obj What if the interval is instead \([1,3/2]\)? (answer), Ex 11.2.5 Compute \(\sum_{n=0}^\infty {3\over 2^n}+ {4\over 5^n}\). Which of the following sequences is NOT a geometric sequence? /Filter /FlateDecode Contact us by phone at (877)266-4919, or by mail at 100ViewStreet#202, MountainView, CA94041. Psychological Research & Experimental Design, All Teacher Certification Test Prep Courses. Other sets by this creator. For each of the following series, determine which convergence test is the best to use and explain why. 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. 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 examine Geometric Series, Telescoping Series, and Harmonic Series. (answer), Ex 11.2.8 Compute \(\sum_{n=1}^\infty \left({3\over 5}\right)^n\). To use integration by parts in Calculus, follow these steps: Decompose the entire integral (including dx) into two factors. Calculus II-Sequences and Series. Which one of these sequences is a finite sequence? Determine whether the series converge or diverge. 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 All other trademarks and copyrights are the property of their respective owners. Good luck! SAT Practice Questions- All Maths; SAT Practice Test Questions- Reading , Writing and Language; KS 1-2 Math, Science and SAT . Sequences can be thought of as functions whose domain is the set of integers. 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 . (answer), Ex 11.4.6 Approximate \(\sum_{n=1}^\infty (-1)^{n-1}{1\over n^4}\) to two decimal places. (answer), Ex 11.2.6 Compute \(\sum_{n=0}^\infty {4^{n+1}\over 5^n}\). /LastChar 127 If it converges, compute the limit. endstream Therefore the radius of convergence is R = , and the interval of convergence is ( - , ). << 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 . Ex 11.7.2 Compute \(\lim_{n\to\infty} |a_{n+1}/a_n|\) for the series \(\sum 1/n\). endobj Alternating Series Test In this section we will discuss using the Alternating Series Test to determine if an infinite series converges or diverges. Each term is the difference of the previous two terms. Here are a set of practice problems for the Series and Sequences chapter of the Calculus II notes. 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 also give many of the basic facts, properties and ways we can use to manipulate a 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, 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. Khan Academy is a nonprofit with the mission of providing a free, world-class education for anyone, anywhere. /FontDescriptor 23 0 R We will focus on the basic terminology, limits of sequences and convergence of sequences in this section. At this time, I do not offer pdfs for solutions to individual problems. n = 1 n2 + 2n n3 + 3n2 + 1. 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 /LastChar 127 /Filter /FlateDecode Donate or volunteer today! We will determine if a sequence in an increasing sequence or a decreasing sequence and hence if it is a monotonic sequence. in calculus coursesincluding Calculus, Calculus II, Calculus III, AP Calculus and Precalculus. Ex 11.7.4 Compute \(\lim_{n\to\infty} |a_n|^{1/n}\) for the series \(\sum 1/n\). Remark. /Type/Font If the series is an alternating series, determine whether it converges absolutely, converges conditionally, or diverges. /Type/Font /BaseFont/SFGTRF+CMSL12 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. /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 The chapter headings refer to Calculus, Sixth Edition by Hughes-Hallett et al. (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 \(\sum_{n=0}^N {1\over e^n} + 10^{-4}\). << Then click 'Next Question' to answer the . )Ltgx?^eaT'&+n+hN4*D^UR!8UY@>LqS%@Cp/-12##DR}miBw6"ja+WjU${IH$5j!j-I1 /LastChar 127 17 0 obj 1 2 + 1 4 + 1 8 + = n=1 1 2n = 1 We will need to be careful, but it turns out that we can . Accessibility StatementFor more information contact us atinfo@libretexts.org. 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 11.11.4 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. 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 Choose your answer to the question and click 'Continue' to see how you did. << Integral Test: If a n = f ( n), where f ( x) is a non-negative non-increasing function, then. We will also give the Divergence Test for series in this section. (b) 555.6 577.8 577.8 597.2 597.2 736.1 736.1 527.8 527.8 583.3 583.3 583.3 583.3 750 %PDF-1.2 )^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 \(\ln 2\). Which of the sequences below has the recursive rule where each number is the previous number times 2? 1. 4 avwo/MpLv)
_C>5p*)i=^m7eE. 5.3.3 Estimate the value of a series by finding bounds on its remainder term. 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? /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 endstream
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Series The Basics In this section we will formally define an infinite series. Which of the following is the 14th term of the sequence below? 777.8 444.4 444.4 444.4 611.1 777.8 777.8 777.8 777.8] These are homework exercises to accompany David Guichard's "General Calculus" Textmap. /FirstChar 0 Ex 11.1.2 Use the squeeze theorem to show that limn n! 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. Harmonic series and p-series. << A brick wall has 60 bricks in the first row, but each row has 3 fewer bricks than the previous one. /FirstChar 0 (answer). S.QBt'(d|/"XH4!qbnEriHX)Gs2qN/G jq8$$< Then determine if the series converges or diverges. 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. Which of the following sequences is NOT a geometric sequence? stream >> Comparison Test: This applies . 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. (a) $\sum_{n=1}^{\infty} \frac{(-1)^n}{\sqrt{n}}$ (b) $\sum_{n=1}^{\infty}(-1)^n \frac{n}{2 n-1}$ 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 The following is a list of worksheets and other materials related to Math 129 at the UA. Chapter 10 : Series and Sequences. About this unit. Determine whether the series is convergent or divergent. (answer), Ex 11.2.2 Explain why \(\sum_{n=1}^\infty {5\over 2^{1/n}+14}\) diverges. /FontDescriptor 8 0 R 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 If you're seeing this message, it means we're having trouble loading external resources on our website. Ex 11.7.3 Compute \(\lim_{n\to\infty} |a_n|^{1/n}\) for the series \(\sum 1/n^2\). (answer), Ex 11.1.4 Determine whether \(\left\{{n^2+1\over (n+1)^2}\right\}_{n=0}^{\infty}\) converges or diverges. Then we can say that the series diverges without having to do any extra work. /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 << 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 (answer). Complementary General calculus exercises can be found for other Textmaps and can be accessed here. Each review chapter is packed with equations, formulas, and examples with solutions, so you can study smarter and score a 5! /LastChar 127 Learn how this is possible, how we can tell whether a series converges, and how we can explore convergence in Taylor and Maclaurin series. The steps are terms in the sequence. Calculus 2. >> /Length 200 endobj 18 0 obj 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 The sum of the steps forms an innite series, the topic of Section 10.2 and the rest of Chapter 10. }\) (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! 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 copyright 2003-2023 Study.com. 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 \(\sum_{n=1}^N {1\over n^4} + 0.005\). ,vEmO8/OuNVRaLPqB.*l. Then click 'Next Question' to answer the next question. Martha_Austin Teacher. Calculus (single and multi-variable) Ordinary Differential equations (upto 2nd order linear with Laplace transforms, including Dirac Delta functions and Fourier Series. UcTIjeB#vog-TM'FaTzG(:k-BNQmbj}'?^h<=XgS/]o4Ilv%Jm The Alternating Series Test can be used only if the terms of the /BaseFont/UNJAYZ+CMR12 Applications of Series In this section we will take a quick look at a couple of applications of series. /Filter[/FlateDecode] All rights reserved. After each bounce, the ball reaches a height that is 2/3 of the height from which it previously fell. 26 0 obj 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. 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 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@ "
8 0 obj 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 endobj Then click 'Next Question' to answer the next question. Your instructor might use some of these in class. Note that some sections will have more problems than others and some will have more or less of a variety of problems. /Name/F1 /Length 1247 Learn how this is possible, how we can tell whether a series converges, and how we can explore convergence in . A review of all series tests. 722.6 693.1 833.5 795.8 382.6 545.5 825.4 663.6 972.9 795.8 826.4 722.6 826.4 781.6 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. 883.8 992.6 761.6 272 272 489.6 816 489.6 816 761.6 272 380.8 380.8 489.6 761.6 272 441.3 461.2 353.6 557.3 473.4 699.9 556.4 477.4 454.9 312.5 377.9 623.4 489.6 272] xYKs6W(MCG:9iIO=(lkFRI$x$AMN/"
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cXf9o/r.&Lxy%/D-Yt+"LX]Sfp]Xl-aM_[6(*~mQbh*28AjZx0 =||. (answer), Ex 11.2.4 Compute \(\sum_{n=0}^\infty {4\over (-3)^n}- {3\over 3^n}\). /Type/Font Complementary General calculus exercises can be found for other Textmaps and can be accessed here. What is the 83rd term of the sequence 91, 87, 83, 79, ( = a. 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. endobj /Subtype/Type1 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 /FirstChar 0 590.3 767.4 795.8 795.8 1091 795.8 795.8 649.3 295.1 531.3 295.1 531.3 295.1 295.1 << Worksheets. (answer), Ex 11.2.3 Explain why \(\sum_{n=1}^\infty {3\over n}\) diverges. We also derive some well known formulas for Taylor series of \({\bf e}^{x}\) , \(\cos(x)\) and \(\sin(x)\) around \(x=0\). It turns out the answer is no. 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! Course summary; . Contact us by phone at (877)266-4919, or by mail at 100ViewStreet#202, MountainView, CA94041. (answer), Ex 11.1.5 Determine whether \(\left\{{n+47\over\sqrt{n^2+3n}}\right\}_{n=1}^{\infty}\) converges or diverges. Sequences and Numerical series.
(answer), Ex 11.9.2 Find a power series representation for \(1/(1-x)^2\). 777.8 777.8] >> Parametric equations, polar coordinates, and vector-valued functions Calculator-active practice: Parametric equations, polar coordinates, . 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. (You may want to use Sage or a similar aid.) (answer). (answer), Ex 11.1.6 Determine whether \(\left\{{2^n\over n! 31 terms. /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 n = 1 n 2 + 2 n n 3 + 3 n . Ratio Test In this section we will discuss using the Ratio Test to determine if an infinite series converges absolutely or diverges. If a geometric series begins with the following term, what would the next term be? 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 . stream In exercises 3 and 4, do not attempt to determine whether the endpoints are in the interval of convergence. >> . 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 >> 611.8 897.2 734 761.6 666.2 761.6 720.6 544 707.2 734 734 1006 734 734 598.4 272 (answer), Ex 11.2.1 Explain why \(\sum_{n=1}^\infty {n^2\over 2n^2+1}\) diverges. If L = 1, then the test is inconclusive. What is the sum of all the even integers from 2 to 250? Psychological Research & Experimental Design, All Teacher Certification Test Prep Courses. /Type/Font /Name/F2 Solution. 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. (answer). nn = 0. Don't all infinite series grow to infinity? 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. Ex 11.9.5 Find a power series representation for \(\int\ln(1-x)\,dx\). When you have completed the free practice test, click 'View Results' to see your results. 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_. We will also briefly discuss how to determine if an infinite series will converge or diverge (a more in depth discussion of this topic will occur in the next section). Infinite series are sums of an infinite number of terms. /Filter /FlateDecode In the previous section, we determined the convergence or divergence of several series by . Integral test. If it converges, compute the limit. 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 >> 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. 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 Root Test In this section we will discuss using the Root Test to determine if an infinite series converges absolutely or diverges. /FontDescriptor 20 0 R endobj /Length 569 1 2, 1 4, 1 8, Sequences of values of this type is the topic of this rst section. Infinite series are sums of an infinite number of terms. Strip out the first 3 terms from the series n=1 2n n2 +1 n = 1 2 n n 2 + 1. 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). /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 We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. (answer), Ex 11.10.9 Use a combination of Maclaurin series and algebraic manipulation to find a series centered at zero for \( x\cos (x^2)\). )^2\over n^n}\) (answer). Comparison tests. 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. Good luck! 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! A proof of the Root Test is also given. %PDF-1.5 Learn for free about math, art, computer programming, economics, physics, chemistry, biology, medicine, finance, history, and more. Then click 'Next Question' to answer the next question. hbbd```b``~"A$"
"Y`L6`RL,-`sA$w64= f[" RLMu;@jAl[`3H^Ne`?$4 (answer). We will also give many of the basic facts and properties well need as we work with sequences. endobj Section 10.3 : Series - Basics. A proof of the Integral Test is also given. More on Sequences In this section we will continue examining sequences. How many bricks are in the 12th row? Choose your answer to the question and click 'Continue' to see how you did. /FirstChar 0 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. Determine whether the following series converge or diverge. { "11.01:_Prelude_to_Sequences_and_Series" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.
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