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WUSTL MATH 132 - 132_09_f14

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Calculus II PLTLFall 2014Worksheet 9These problems are to be do ne without the use of a calculator unless otherwisespecified.1) (Round Robin) Use the Ratio Test to explore the convergence of the seriesP∞n=1n!3n.2) (Pairs) State the conditions under which the Alternating Series Test can be used.Then explain how the test can be used to determine whether a series converges or diverges.(a) Evaluate the convergence or divergence ofP∞n=1(−1)n+12n+1.(b) Explain why the Alternating Series Test cannot be used to testP∞n=1(−1)nn2n+1for convergence.It is a fact that∞Xn=0(−1)nn!=1e.(c) Show that this series satisfies the requirements of the Alternating Series Test.If the Alternating Series Test is applicable to a convergent seriesPan, you can approximatethe exact value of the su m to a specified degree of accuracy by taking a large enoughpartial s um. For example, if the sum of the series , s, is estimated with the partial sums4= a0+ a1+ a2+ a3+ a4, then the absolute value of the error, |s −s4| = |P∞n=5an| mustbe less than | a5|.(d) Explain in your own words why the last statement is true.(e) Verify that |s −s4| ≤ |a5| for this series. Note that s4estimates the value of e−1correctly to two decim a l places.(f) Now estimate the value of 1/e accurate to four decimal pla ces .3) (Scribe) Use the Comparison Test to show that the series converges.∞Xn=1n23n2+ 4n4) (Pairs) Determine whether each of the following series is absolutely convergent,conditionally convergent, or divergent.(a)P∞n=3(−1)nln n(b)P∞n=0(−1)nn32n5) (Round Robin) Determ ine whether each series is convergent or divergent. Make s ureto indicate which tests can be used, and whether any tests are indeterminate for eachseries .(a)P∞n=2(n2− 1)/(n3− n − 1)(b)P∞n=21/(n√ln n)(c)P∞n=03n2n+4n(d)P∞n=11000/√n3+ 2(e)P∞n=0sin n6) ( ) Use an example or a picture to illustrate whether each statement is trueor false.(a) For |r| < 1 and a a real number,P∞n=1arn−1has the same sum a sP∞n=0arn.(b) If a seriesPanis such that the terms antend to zero as n incr eases, then it isstill possible that the series is divergent.(c) IfP|an| converges, thenP(−1)n|an| converges.(d) If the sequence anis not bound ed, and anis po sitive for n > 100, then thesequence has a term greater than 1,000,000.(e) If f(x) > 0 and f is decreasi ng andR∞af(x) dx = K < ∞ for some real numbera, thenP∞n=1f(n) is convergent.(f) If a mon otonic sequence of positive terms does not converge, then it has a termgreater than one g oo g


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WUSTL MATH 132 - 132_09_f14

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