Calculus II PLTL Fall 2014 Worksheet 9 These problems are to be done without the use of a calculator unless otherwise specified P 1 Round Robin Use the Ratio Test to explore the convergence of the series n 1 3n n 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 n 1 P a Evaluate the convergence or divergence of n 1 1 2n 1 P n2 b Explain why the Alternating Series Test cannot be used to test n 1 1 n n 1 for convergence It is a fact that X 1 n 1 n e n 0 c Show that this series satisfies the requirements of the Alternating Series Test P If the Alternating Series Test is applicable to a convergent series an you can approximate the exact value of the sum to a specified degree of accuracy by taking a large enough partial sum For example if the sum of the series s is estimated with the P partial sum s4 a0 a1 a2 a3 a4 then the absolute value of the error s s4 n 5 an must be 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 s4 estimates the value of e 1 correctly to two decimal places f Now estimate the value of 1 e accurate to four decimal places 3 Scribe Use the Comparison Test to show that the series converges X n 1 n2 3n2 4 n 4 Pairs Determine whether each of the following series is absolutely convergent conditionally convergent or divergent a 1 n n 3 ln n P b 1 n n3 n 0 2n P 5 Round Robin Determine whether each series is convergent or divergent Make sure to indicate which tests can be used and whether any tests are indeterminate for each series P 2 3 a n 2 n 1 n n 1 P b n 2 1 n ln n P 3n c n 0 2n 4n P d 1000 n3 2 n 1 P e n 0 sin n 6 or false Use an example or a picture to illustrate whether each statement is true P P For r 1 and a a real number n 1 arn 1 has the same sum as n 0 arn P b If a series an is such that the terms an tend to zero as n increases then it is still possible that the series is divergent P P c If an converges then 1 n an converges a d If the sequence an is not bounded and an is positive for n 100 then the sequence has a term greater than 1 000 000 R e PIf f x 0 and f is decreasing and a f x dx K for some real number a then n 1 f n is convergent f If a monotonic sequence of positive terms does not converge then it has a term greater than one googol
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