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Name: Math 1220 Exam 4 Review 1. For each of the following sequences: • Determine if the sequence converges • If it converges, find the limit of the sequence as n → ∞ (a) an = n3 + ln(n) (n + 1)3 (b) an = e−n sin(n) (c) 1, 1 1 1 , , ,... 1 2 1 − 2 1 − 3 1 − 34 2. Determine if the following series converge. If the series converges, find its sum. (a) ∞ X 2n 3n+1 (b) ∞ X n=1 n=1 1 n(n + 1) 3. Determine if the following series converge or diverge. Be sure to justify your conclusion and specify what tests you used. (a) ∞ X n2 + 5 n=1 (b) ∞ X n2 − 5 ne−n 2 (c) n=1 (d) n=1 ∞ X (ln(n))2 (e) n n=1 √ ∞ X n+2 n n=1 ∞ X n2 (f) n3 − 6 n! ∞ X 1 + cos2 (n) n=1 n2 4. Determine if the following series converge conditionally, converge absolutely, or diverge. Be sure to justify your conclusion and specify what tests you used. (a) ∞ X n=1 (b) ∞ X n=1 ∞ X 1 (c) (−1)n √ 2 n +1 n=1 1 (−1) n ln n n (−1)n n3 n! (d) ∞ X sin(πn) n=1 n2 5. Determine the error in using the partial sum S4 to approximate the sum of the following series: ∞ X (−1)n n=1 1 n2 + 1 How many terms would you need to sum in order for the error to be less than 0.1? 6. For what values of x does the following series converge? ∞ n X x n=0 2n