Name Section 8 00 9 30 11 00 12 30 Math 1B Quiz 5 SOLUTIONS October 4 2007 GSI Rob Bayer You have twenty minutes to complete this quiz You must show your work 1 3 pts Determine whether the sequence an converges find its value 2 sin n converges or diverges If it n2 ln n n 3 1 n2 sin Since 1 sin n 1 we have n2 ln 2 ln n n2 ln n Both the left and right n hand side of this inequality go to 0 as n so by the squeeze theorem the original sequence must also 2 4 pts Determine whether the series X 2 converges or diverges If it converges 2 n 2 n 1 find its value A B n 1 so 2 A n 1 B n 1 which We start with partial fractions n22 1 n 1 gives the equations A B 0 A B 2 This has A 1 B 1 for a solution so 2 1 1 n 1 n 1 n2 1 Let s try a few partial sums 1 1 1 3 1 1 3 1 1 3 1 1 2 1 1 2 s2 s3 s4 sn So lim sn n 1 1 2 4 1 1 1 1 2 4 3 5 1 1 4 5 1 1 n n 1 3 and the sum is convergent and has value 2 3 2 3 3 pts True False For those that are true give a short explanation of why and for those that are false give a counterexample a If an and bn are sequences such that an and an bn both converge then bn converges True If an and an bn are both convergent then an bn an is convergent but this is just bn b Every bounded sequence converges False an 1 n is one possible counterexample c If sn is the sequence of partial sums for n 1 an and limn sn 0 then converges P True This is actually the definition of an 0 P P n 1 an
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