12-4 - 1 12.4: Examples Determine if the series converges...

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12.4: The Comparison Tests The Comparison Test allows us to determine the con- vergence or divergence of a series a n by comparing it to a series b n we already understand. Theorem (The Comparison Test) . Suppose that a n and b n are series with positive terms. 1. If b n is convergent and a n b n for all n , then a n is also convergent. 2. If b n is divergent and a n b n for all n , then a n is also divergent. The geometric series, p -series, and harmonic series are often good candidates to use when trying to find a series for comparison since we understand the convergence/divergence for each of these series.
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Unformatted text preview: 1 12.4: Examples Determine if the series converges or diverges. 1. ∑ ∞ n =1 1 1+2 n 2. ∑ ∞ n =1 1 √ n 2-1 3. ∑ ∞ n =1 n-1 n 4 n 2 12.4: The Comparison Tests Theorem (The Limit Comparison Test) . Suppose that ∑ a n and ∑ b n are series with positive terms. If lim n →∞ a n b n = c where c is a finite number and c > 0, then either both series converge or both diverge. 3 12.4: Examples Determine if the series converges or diverges. 1. ∑ ∞ n =1 1 2 n-1 2. ∑ ∞ n =1 √ n +2 2 n 2 + n +1 4...
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This note was uploaded on 04/04/2011 for the course MATH 408 L taught by Professor Zheng during the Fall '10 term at University of Texas.

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12-4 - 1 12.4: Examples Determine if the series converges...

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