Calculus II PLTL Fall 2014 Worksheet 8 These problems are to be done without the use of a calculator unless otherwise specified Before beginning to work on the following problems explain what these statements mean 1 2 3 4 The The The The sequence sn n 0 converges sequence sn n 1 is divergent P series n 2 an is convergent P series n 3 an diverges 1 Scribe Find a formula for the nth partial sum of the series it to find the series sum if it converges P 2 n 1 n2 4n 3 and use 2 Round Robin Match each of the general terms from a sequence a e with the correct description of its behavior as n I V a b c d e an 1 5 n bn 1 ln n ln n 1 cn 2 1 n dn 1 n1 n en n 1 n n I II III IV V Diverges oscillates Diverges to Converges to 0 Converges to 1 Converges to 1 e 3 Pairs Let an n 100 2 1000 a Compute the value of each term indicated below a1 a2 a3 a4 a96 a97 a98 a99 Are the terms increasing or decreasing Do the terms seem to be tending to a limit b Now compute the values of these terms a101 a102 a103 a104 What is going on here Does this sequence seem to be converging now c Compute limn an Does this tell you whether an converges or diverges d Let f x x 100 2 1000 Then an f n for all positive integers n Sketch the graph of f How does this help to explain the answers you got in a c e This example shows that it can be useful to look at a real valued function to understand the behavior of a sequence One way that the behavior of functions that match up with terms of series at integer values can sometimes be determined is with the Integral Test which can t be applied to this series Why not 4 Scribe Determine whether the series P 2 n 3 1 n ln n ln ln n converges or diverges 5 Use a geometric series to rewrite the repeating decimal 3 142857142857142857 as a rational number P 6 Pairs Suppose that an converges with an 0 for all n Decide if each of the following series converges diverges or if there is not enough information provided P a an n P b 1 an P c nan P d an an 2 P e an 2
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