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Unformatted text preview: Homework #6 14.3 Determine which of the following series converge. Justify your answers. (a) 1 / n ! (b) (2 + cos n ) / 3 n (c) 1 / (2 n + n ) (d) (1 / 2) n (50 + 2 /n ) (e) sin( n/ 9) (f) (100) n /n ! Solution: (a): Applying the Ratio Test, we obtain limsup a n +1 a n = limsup 1 / p ( n + 1)! 1 / n ! = limsup s n ! ( n + 1)! = limsup 1 n + 1 = r lim 1 n + 1 = 0 = 0 < 1. Hence, the series converges. (b): Since 0 2 + cos n 3 n 3 3 n for all n , and since the geometric series X 3 3 n = X 3 1 3 n converges, the Comparison Test tells us that the given series converges. (c): Again we use comparison. Since 0 1 2 n + n 1 2 n for all n , and since the geometric series X 1 2 n = X 1 2 n converges, the Comparison Test says the given series converges. (d): Since 0 50 + 2 n 2 n 52 2 n , and since the geometric series X 52 2 n = X 52 1 2 n converges, the Comparison Test tells us that the given series converges. (e): Notice that the sequence ( sin( n/ 9) ) diverges. In particular, it does not converge to zero. By Corollary 14.5, the given series diverges. (f): Applying the Ratio Test, we obtain limsup (100) n +1 / ( n + 1)!...
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 Spring '09

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