Math107_ExRev_Series_Ans - Math 107 Extra Review for Final...

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Math 107 Extra Review for Final over Series Answers 1. ln 1 k " # $ % & k = 1 ( ) : Use n th -Term Test, limit k " # ln 1 k $ % & ( ) = ln 1 " # $ % & ( = ln 0 ( ) = "# . Since the limit 0, then by the n th -Term Test the series ln 1 k " # $ % & k = 1 ( ) diverges . 2. 2 10 k k = 1 " # : Use Geometric Series, this is a geometric series with r = 1 10 , since this is less than 1, then the series converges (absolutely). 3. k k + 1 k = 1 " # : Use n th -Term Test, limit k " # k k + 1 , divide all terms by k to get limit k " # 1 1 + 1 k = 1 1 + 1 " = 1 1 + 0 = 1 . Since the limit 0, then by the n th -Term Test the series k k + 1 k = 1 " # diverges . 4. 1 ln ln k ( ) k = 3 " # : Use Direct Comparison Test, start with ln k < k for all k > 3, this implies ln ln k ( ) < k which leads to 1 ln ln k ( ) > 1 k . The series 1 k k = 3 " # is the harmonic series and diverges. Since 1 k k = 3 " # is a smaller divergent series, then the larger series 1 ln ln k ( ) k = 3 " # must also diverge . 5. ln k ( ) 2 k 3 2 k = 1 " # : Use Limit Comparison Test with 1 k 5 4 k = 1 " # . Use 1 k 5 4 because 1 k 3 2 " 1 k 5 4 " 1 k 1 . 1 k 5 4 k = 1 " # is a p-series with p = 5 4 which is greater than 1, therefore 1 k 5 4 k = 1 " # converges. Next, limit k " # ln k ( ) 2 k 3 2 1 k 5 4 = limit k " # ln k ( ) 2 k 3 2 $ k 5 4 1 = limit k " # ln k ( ) 2

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