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sec11_6

# sec11_6 - 3 3 Root Test Theorem 3.1 Let ∑ a n be a series...

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11.6 Absolute Convergence and the Ratio and Root Test 1. Absolute Convergence Defnition 1.1. A series a n is called absolutely convergent if the series | a n | converges. Example 1.1. ± n =1 ( - 1) n n 2 Remarks 1.1. (1) -| a n |≤ a n ≤| a n | ,so 0 a n + | a n |≤ 2 | a n | (2) If ± | a n | converges, then . (3) If ± a n converges, then . Defnition 1.2. A series a n is called conditionally convergent if the series a n converges, but the series | a n | diverges. Example 1.2. (problem 12) Determine if the series is absolutely convergent, condi- tionally convergent or divergent. ± n =1 sin 4 n 4 n 1

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Section 11.6 Absolute Convergence and the Ratio and Root Tests 2 2. Ratio Test Theorem 2.1. Let a n be a series with lim n →∞ ± ± ± ± a n +1 a n ± ± ± ± = L ( L is not necessarily a real number) (1) If 0 L< 1 , then (2) If L> 1 or L = , then (3) If L =1 , then Example 2.1. (problem 18) Determine if the series is absolutely convergent, condi- tionally convergent or neither. ² n =1 n ! n n
Section 11.6 Absolute Convergence and the Ratio and Root Tests

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Unformatted text preview: 3 3. Root Test Theorem 3.1. Let ∑ a n be a series with lim n →∞ n ± | a n | = L ( L is not necessarily a real number) (1) If ≤ L < 1 , then (2) If L > 1 or L = ∞ , then (3) If L = 1 , then Example 3.1. (problem 20) Determine if the series is absolutely convergent, condi-tionally convergent or neither. ∞ ² n =1 (-2) n n n Section 11.6 Absolute Convergence and the Ratio and Root Tests 4 4. Varied Examples Example 4.1. (problem 14) Determine if the series is absolutely convergent, condi-tionally convergent or neither. ∞ ± n =1 (-1) n +1 n 2 2 n n ! Example 4.2. (problem 22) Determine if the series is absolutely convergent, condi-tionally convergent or neither. ∞ ± n =2 ²-2 n n + 1 ³ 5 n...
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sec11_6 - 3 3 Root Test Theorem 3.1 Let ∑ a n be a series...

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