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Unformatted text preview: vergence , although it does aﬀect what the series converges to .) ± example 1.2. Does the the ratio test applies to the series ∞ X k =1 x k k ! 1 solution. We have a k +1 a k = x k +1 ( k +1)! x k k ! = x k + 1 → 0 as k → ∞ Thus r = 0 and we have convergence for every x . example 1.3. Does the the ratio test applies to the series ∞ X k =1 1 2 k +(1) k solution. We have a k +1 a k = 1 2 k +1+(1) k +1 1 2 k +(1) k = 2 k +(1) k 2 k +1+(1) k +1 = 1 2 · 2 2(1) k = ± 2 if k is even 1 8 if k is odd Thus lim k → ∞ a k +1 a k doesn’t exist, and the test is inconclusive. 2...
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 Fall '08
 HIMONAS
 Calculus, Mathematical Series, 1 K, 1 2K, ak Ak

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