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Absolute Convergence Ratio Test Root Test
Lesson 11-6 Absolute Convergence Ratio Test Root Test
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Vocabulary Absolute Convergent – a series of numbers that alternate in sign, like the summation of the following Conditionally Convergent -- a convergent series that is not absolutely convergent Rearrangement – a reordering of terms in an infinite series
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Types of Series Geometric Telescoping Harmonic P-Series Alternating
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Types of Series Tests Divergence (easiest to check)
if lim an ≠ 0 then it diverges if lim an = 0 then it is inconclusive Comparison (last resort) if an ≤ bn for all n 0 then it diverges Limit Comparison (better than comparison) Ratio (apply when factorials are involved) Root (apply when nth power involved) n→∞
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Ratio Test | | | | Let ∑ an be a series
If , then the series ∑an is absolutely convergent (and therefore convergent) If , or , then the series ∑an is divergent If , the Ratio test is inconclusive (no conclusion about the convergence or divergence of the series can be drawn) an+1 lim = L < 1 an | n→∞ an+1 lim = L > 1 an | n→∞ an+1 lim = ∞ an n→∞ | an+1 lim = 1 an n→∞ |
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Root Test Let ∑ an be a series
If , then the series ∑an is absolutely convergent (and therefore convergent) If , or , then the series ∑an is divergent If , the Ratio test is inconclusive (no conclusion about the convergence or divergence of the series can be drawn) lim = L < 1 n→∞ √|an| n lim = L > 1 n→∞ √|an| n lim = ∞ n→∞ √|an| n lim = 1 n→∞ √|an| n
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11-6 Example 1 | Is the following series convergent or divergent?
(-1)n+1 2n series n! Lim bn = indeterminate and L’Hospital’s Rule does not apply! n→∞ Use Ratio Test, the absolute values eliminate the (-1) term and we get an n+1 / (n+1)! lim = lim an n / n! | n→∞ n! 2n This leads to lim = = 0 therefore it converges (n+1)! 2n n+1 7
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11-6 Example 2 Is the following series convergent or divergent? 6n
Lim = indeterminate and L’Hospital’s Rule does not apply! n→∞ 6n n - 1 n Ratio test is impossible, so we can try the Root test Lim = Lim = 2 n→∞ 6n n - 1 n 1/n since p > 1 the series diverges by the Root Test 8
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Homework Pg : problems 3, 4, 7, 11
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