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Fall 2002 Infinite Series Problems answers

Thomas' Calculus: Early Transcendentals

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Mat104 Fall 2002, Infinite Series Problems From Old Exams For the following series, state whether they are convergent or divergent, and give your reasons. (1) 1 n , diverges by the limit comparison test (LCT) (2) converges by ratio test (3) converges by ratio test (4) 1 n , diverges by LCT (5) 2 n 3 n , converges by LCT (6) converges by the alternating series test (AST). It is conditionally convergent only since taking absolute values gives a divergent sum. (ln n < n implies that ln(ln n ) < ln n so 1 ln(ln n ) > 1 ln n > 1 n ) (7) converges by AST, conditionally convergent since summing 1 / n gives a divergent series (p-test with p = 1/2). (8) convergent by LCT (9) 1 n 3 so convergent by LCT (10) diverges since a n → ∞ as n → ∞ . (11) 1 n so divergent by LCT (12) convergent by the ratio test (13) conditionally convergent (14) converges by ratio test. a n +1 /a n 1 /e . (15) converges by the nth root test. (16) convergent by LCT. Asymptotic to 2 n + 6 n 7 n , the sum of two convergent geometric series.
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