Math 1B PDP Worksheet Rob Bayer March 17 2009 Integral Test 1 Determine whether each of the following series converge or diverge DO NOT use the series comparison test X 3 n a n n 1 b 1 X c 1 16 1 81 1 256 2 x 9 x6 n 1 d X 1 n 3 n ln n ln ln n 2 For which values of x does X ln x n converge And for n 1 3 Consider the series X n 5 3 X ln n x n 1 1 Re index this series to turn it into a p series and determine if it converges or n 3 diverges If you finish either section early move on to the crazy facts section below Comparison Test 1 Determine whether each of the following series converge or diverge a b c d X n 3 3 4n 2 n n 1 X n 4 X n 2 X 2n e X 1 n n2 n 1 X ln n n n 2 2 2 X n 2 g n3 5 n 3 en ln n f sin2 n ln n n2 1 n en n 1 P P 2 2 a Show that if an is a convergent series with non negative terms then an is also convergent P P b However if an is a convergent series with non negative terms an could either converge or diverge Give an example for each of these possibilities P P P c Show that if an and bn are both convergent series with positive terms then an bn converges too Crazy Facts 1 Find the flaw in the following proof that 0 1 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 1 2 The Cantor Set is a set of real numbers constructed as follows start with the interval 0 1 and remove the middle third of it That is remove the interval 31 23 leaving 0 31 23 1 Now remove the middle third of each of these remaining intervals leaving 0 19 29 13 23 79 89 1 After continuing this process infinitely many times you will be left with the Cantor set a Show that the total length of all the intervals you remove is 1 b Convince yourselves that despite this the cantor set has infinitely many numbers in it Give some examples of these numbers c Side note it actually turns out that the Cantor Set is uncountable meaning there are exactly the same number of numbers in it as there were in the interval 0 1 before you started removing middle thirds The proof of this is actually very easy but requires some knowledge of binary and ternary decimal systems Talk to me if you re curious 3 Recall that the Fibonacci sequence is defined as F1 1 F2 1 Fn Fn 1 Fn 2 a If you haven t before write out a few terms of this sequence to get a sense of the pattern b Show that c Show that d Show that 1 Fn 1 Fn 1 1 Fn 1 Fn X 1 1 F F n 2 n 1 n 1 X Fn 2 F F n 2 n 1 n 1 1 Fn Fn 1
View Full Document
Unlocking...