Unformatted text preview: ∑ a n is a series with positive terms. (a) Let L = max ± lim sup n →∞ a 2 n a n , lim sup n →∞ a 2 n +1 a n ² and l = min ± lim inf n →∞ a 2 n a n , lim inf n →∞ a 2 n +1 a n ² . Then 1. If L < 1 2 then ∑ a n converges. 2. If l > 1 2 then ∑ a n diverges. 3. If l ≤ 1 2 ≤ L then the test is inconclusive 1 Hint: Write ∑ m k = n a k ≤ ∑ ∞ k = n S k , with S k = a 2 k n + a 2 k n +1 + ··· + a 2 k +1 n1 . (b) Use this test to show that the series ∑ a n with a n = (2 n1)!! 2 n ( n +1)! (for which both the ratio and root tests fail) converges 2 . 1 IE, there exist both convergent and divergent series for which this condition holds. 2 Traditionally, n !! does not mean ( n !)!. Rather, it usually means n ( n2)( n4) ··· 1. 1...
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 Spring '11
 Peters
 Numerical digit, Numeral system, Countable set

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