Unformatted text preview: Math 152 Workshop 12 Spring 2011 1. Under the hypotheses of the integral test, if a n = f ( n ) and if s n = a 1 + a 2 + Â·Â·Â· + a n = n âˆ‘ j =1 a j , then R n 1 f ( x ) dx â‰¤ s n â‰¤ a 1 + R n 1 f ( x ) dx for each positive integer n . For the harmonic series âˆž âˆ‘ j =1 1 j , this implies ln n â‰¤ 1 + 1 2 + 1 3 + Â·Â·Â· + 1 n â‰¤ 1 + ln n for each positive integer n . (a) Find the analogous inequalities for the series âˆž âˆ‘ j =1 1 âˆš j and for the series âˆž âˆ‘ j =2 1 j ln j . (b) Estimate the sum of the first 10 10 terms of the series, in each of the three cases. Then estimate the sum of the first 10 100 terms. (c) Of the three series, which diverges the fastest? the slowest? 2. Under the hypotheses of the integral test, if a n = f ( n ) then for any positive integer N , âˆž âˆ‘ N +1 a n â‰¤ âˆž R N f ( x ) dx . (a) How large does N have to be to ensure that (i) N âˆ‘ n =1 1 n 5 is within 10 6 of âˆž âˆ‘ n =1 1 n 5 ?...
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 Spring '11
 SOSA
 Harmonic Series

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