This question concerns the following exercise. Exercise Determine whether the sequence (an) given by 312-2 3n2-1...

  

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This question concerns the following exercise. Exercise Determine whether the sequence (an) given by 312-2 3n2-1 converges, and find its limit if it does. Name any results or rules that you use. You may use the basic null sequences listed in Theorem D7 from Unit D2. (a) Explain why the following solution to this exercise is incorrect and/or incomplete, identifying three errors or significant omissions. For each error or omission, explain the mistake that the writer of the solution has made. (There may be more than three errors or omissions, but you need identify only three. These should not include statements or omissions that follows logically from earlier errors or omissions.) Solution (incorrect and/or incomplete!) For n = 1,2,..., we have lanl 3n2 2 312-1 VI + 3n2-2 312 (by the Triangle Inequality) + 3n2-2 312-1 (since >0 and 3n2 -2 3n2 > 0) -1 113 + 3n2 3n2 2n 3 Now 2n/3oo as noo, so by the Squeeze Rule (Theorem D18) it follows that laloo as n→oc. Therefore an oo as n→ oo. Thus (a) is divergent. (b) Write out a correct solution to the exercise. Theorem D7 Basic null sequences The following sequences are null. (a) (1/nº), for p > 0. (b) (c), for c< 1. (c) (n'c"), for p > 0, |c| < 1. (d) (c/n!), for cЄ R. (e) (nº/n!), for p > 0. This question concerns the following exercise. Exercise Determine whether the sequence (an) given by 312-2 3n2-1 converges, and find its limit if it does. Name any results or rules that you use. You may use the basic null sequences listed in Theorem D7 from Unit D2. (a) Explain why the following solution to this exercise is incorrect and/or incomplete, identifying three errors or significant omissions. For each error or omission, explain the mistake that the writer of the solution has made. (There may be more than three errors or omissions, but you need identify only three. These should not include statements or omissions that follows logically from earlier errors or omissions.) Solution (incorrect and/or incomplete!) For n = 1,2,..., we have lanl 3n2 2 312-1 VI + 3n2-2 312 (by the Triangle Inequality) + 3n2-2 312-1 (since >0 and 3n2 -2 3n2 > 0) -1 113 + 3n2 3n2 2n 3 Now 2n/3oo as noo, so by the Squeeze Rule (Theorem D18) it follows that laloo as n→oc. Therefore an oo as n→ oo. Thus (a) is divergent. (b) Write out a correct solution to the exercise. Theorem D7 Basic null sequences The following sequences are null. (a) (1/nº), for p > 0. (b) (c), for c< 1. (c) (n'c"), for p > 0, |c| < 1. (d) (c/n!), for cЄ R. (e) (nº/n!), for p > 0.