Math 1B Section 107 Quiz 7 Thursday 11 October 2007 GSI Theo Johnson Freyd http math berkeley edu theojf Name 1 True or False 1 pt each For each of the following statements decide if it is true or false You do not need to show work I will grade only your answers P P a Let s say 0 an bn and n 1 bn B If A B then n 1 an A and an bn for every n P TRUE The first inequality guarantees that bn an B A 0 is a sum of nonnegative terms but the only way to add up nonnegative numbers and get 0 is if each of the numbers is itself 0 b If 0 an f n where f x is a continuous decreasing function on x 1 R P a such that 1 f x dx converges then n 1 n converges TRUE This is a combination of the comparison and integral tests P P c If 0 an 2 and n 1 sin an diverges then n 1 an diverges TRUE This follows from the comparison test since sin x x if x 0 and so an sin an 1 For the next two questions use either the Limit Comparison Test or the Integral Test to determine if the series converges or diverges Be sure to check that the series satisfies the conditions necessary for the test 2 3 pts X arctan n n2 ln n n 1 We recall that limn arctan n 2 Then comparing with 1 n2 which converges gives arctan n n2 ln n 1 lim arctan n 1 2 2 n n 1 n 1 ln n n 2 P which is more than 0 and less than So since 1 n2 converges so must our series P arctan n n2 ln n lim 3 4 pts X n 3 1 n ln n ln ln n Recognizing this as something integrable we check the conditions for the integral test 1 x ln x ln ln x is always positive It is the product of three decreasing functions 1 x 1 ln x and 1 ln ln x so must itself be decreasing Thus using the integral test we check Z Z dx du x ln x ln ln x 3 ln ln 3 u where we substitute u ln ln x Thus the original series also diverges 2
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