workshop9 - n =2-1 n n(ln n 2 converges A computer gives...

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Workshop 9 1) Under the hypotheses of the integral test, if a n = f ( n ) and if s n = a 1 + a 2 + · · · + a n = n j =1 a j , then n 1 f ( x ) dx s n a 1 + n 1 f ( x ) dx for each positive integer n . For the harmonic series j =1 1 j , this implies ln n 1+ 1 2 + 1 3 + · · · + 1 n 1+ln n for each positive integer n . a) Find the analogous inequalities for the series j =1 1 j and for the series 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 , N +1 a n N f ( x ) dx . a) How large does N have to be to ensure that (1) N n =1 1 n 5 is within 10 - 6 of n =1 1 n 5 ? (2) N n =1 ne - n 2 is within 10 - 6 of n =1 ne - n 2 ? b) Get a decimal approximation for the sum of one of the series with error less than 10 - 6 . 3) a) Verify that the infinite series
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Unformatted text preview: n =2 (-1) n n (ln( n )) 2 converges. A computer gives the approximate value .84776 to 5 digit accuracy for the sum of this series. Find a specic partial sum which is guaranteed to give this number to 5 digit accuracy. Give evidence supporting your assertion. b) Verify that the innite series n =2 1 n ln( n ) diverges. A computer gives the approximate value of 4.74561 for the 10 , 000 th partial sum. Are the partial 1 sums of this series unbounded? If yes, nd a specic partial sum which is guaranteed to be greater than 100. Give evidence supporting your assertion. Comment In neither case is the best possible partial sum requested. Sup-porting evidence must be presented for the two partial sums given. 2...
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