Math 1B PDP Worksheet Rob Bayer March 10 2009 More Sequences 1 Determine whether each of the following sequences are monotone bounded both or neither 1 3n 4 1 b sin n a c 3n 2 n 1 2 Give an example of each of the following a A bounded sequence that diverges b A monotone sequence that diverges c A convergent sequence that isn t monotone 3 In addition to defining sequences by a formula of the form an f n we can also define a sequence by giving the value of a1 and a rule for getting the next term in the sequence from the previous one For example we could say an 3 an 1 a1 2 This then defines the sequence a1 2 a2 3 a1 3 2 5 a3 3 a2 8 a4 3 a3 11 etc Such a definition is called recursive A definition that doesn t refer to previous terms or have any s in it is called a closed form For example a closed form for the sequence above would be an 2 3 n 1 For each of the following recursively defined sequences find a closed form for an Use your answer to determine if each is convergent or divergent a an nan 1 a1 1 b an an 1 n a1 1 c an ran 1 a1 a where a r are contants and r 1 d an ln n an 1 a1 0 Series I 1 Re write each of the following in P notation Start your sum wherever is convenient a 1 2 3 b 1 1 4 1 9 c 1 1 1 1 d 3 2 3 4 3 8 3 16 2 Re index each of the following series to start at n 0 a X arn 1 b n 1 X ln n 2 n 1 n 2 c X sinn n 1 n 2 3 Determine whether each of the following series are convergent or divergent For those that are convergent find the sum a X 3n 2 22n n 1 b X n 2 c n 1 X 1 n n 0 2 3n 4n d X n 1 n2 n2 3 3n 1 4 By using partial sums ie the definition of a series determine whether each of the following converge or diverge For those that converge find the sum a X sin n 1 sin n b n 1 5 If the nth partial sum of a series X ln n 2 X n 1 an is sn n2 2n 1 n2 c X n 1 find an and an n 1 n 1 6 For which values of x do each of the following converge X 2 n n 2 n 1 a b X n 0 X n 0 xn c X x n n 1 xn 2 n d X x 1 n 32n 1 n 0 7 True false For those that are true provide a brief explanation intuition of why For those that are false find a counterexample P a If an is positive for all n and each partial sum is less than 104 then n 0 an converges b If an bn for all n and both sequences converge then lim an lim bn c If sn is the sequence of partial sums for the sequence an and limn an 0 then limn sn exists
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