Calc09_5 - 9.5 Testing Convergence at Endpoints The...

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9.5 Testing Convergence at Endpoints Greg Kelly, Hanford High School, Richland, Washington The original Hanford High School, Hanford, Washington
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Remember: The series converges if . 1 L < The series diverges if . 1 L The test is inconclusive if . 1 L = The Ratio Test : If is a series with positive terms and n a 1 lim n n n a L a + →∞ = then:
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This section in the book presents several other tests or techniques to test for convergence, and discusses some specific convergent and divergent series.
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The series converges if . 1 L < The series diverges if . 1 L The test is inconclusive if . 1 L = Nth Root Test : If is a series with positive terms and n a lim n n n a L →∞ = then: Note that the rules are the same as for the Ratio Test.
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example: 2 1 2 n n n = 2 2 n n n 2 2 n n = 2 lim n n n →∞ ( 29 2 lim n n n →∞ = ?
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lim n n n →∞ 1 lim n n n →∞ ( 29 1 lim ln n n n e →∞ 1 lim ln n n n e →∞ ln lim n n n e →∞ 1 lim 1 n n e →∞ 0 e 1 Indeterminate, so we use L’Hôpital’s Rule formula #104 formula #103
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example: 2 1 2 n n n = 2 2 n n n 2 2 n n = 2 lim 2 n n n →∞ 2 lim n n n →∞ ( 29 2 lim n n n →∞ = 2 1 = 1 = 1 2 = it converges ?
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