Unformatted text preview: ³ ≤ n k 2 n = c n ( √ 2)n . We know that c n → 0 as n → ∞ (see the previous exercise sheet). Therefore, by ratio test, the series ∑ ∞ n =1 c n ( √ 2)n converges. Now the comparison theorem implies that the series ∑ ∞ n =1 (1) n n k 1+2 n is also convergent. 5(iv). We have lim n1 n +1 = 1. The ratio test implies that the series converges for all x such that  x + 1  < 2. If  x + 1  ≥ 2 then the sequence ( n1) ( x +1) n ( n +1) 2 n does not converge to 0 and, consequently, the series diverges. 1...
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 Spring '09
 Calculus, Trigraph, lim N→∞

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