Unformatted text preview: n n 3 (or 1 n 2 ) and that the series is likely convergent. To ﬁnd an upper bound b n , we bound the numerator above and the denominator below as follows: For all n ≥ 1, ln n < n and n 3 + 1 > n 3 from which it follows that 1 n 3 + 1 < 1 n 3 . Thus ln n n 3 + 1 < n n 3 = 1 n 2 . The series ∞ X n =1 1 n 2 is a convergent p –series, therefore by the Direct Comparison Test, both series converge....
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 Fall '10
 KIHYUNHYUN
 Calculus, Mathematical Series, Convergence, Dominated convergence theorem

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