1120_11 - Week 11 Outline The Integral Test p-series 1 np...

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Week 11 Outline The Integral Test p -series 1 n p –Estimate the sum of a series Comparison test Limit Comparison Test Ratio Test Root Test Strategy for testing series Theorem Suppose that f is a continuous, positive, nonincreasing function on [1 , ) and a n = f ( n ). Then a n is convergent if and only if R 1 f ( x )d x is convergent. In other words, (1) R 1 f ( x ) d x is convergent then a n is convergent. (2) R 1 f ( x ) d x is divergent then a n is divergent. Remarks 1. It is not necessary to start the series or the integral at n = 1. For example, since R N d x x 2 is convergent, we conclude that k = N a k is convergent and hence so is k =1 a k . 2. We CANNOT remove the assumption that f is nonincreasing. For example, let f ( x ) = | sin( πx ) | . Then R 1 f ( x )d x diverges, however a n = f ( n ) = 0 for all n 1. On the other hand, let f ( x ) = ( n, if n - 1 n · 2 n x n 0 , otherwise , then R 1 f ( x )d x n =1 1 2 n = 1, however a n = f ( n ) = n for all n 1. Examples Determine whether the following series are convergent or divergent. 1. Harmonic series 1 n 2. 1 n 2 3. 1 n 4. Find the range of p such that the p -series 1 n p converges. 5. Find the range of p such that the series 1 n (ln n ) p converges 6. Find all b > 0 such that b ln n converges. 1
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7. Find the range of a such that ∑( a n +2 - 1 n +4 ) 8. 1 n 2 +1 9. 1 n ( n +1) 10. n =3 1 n (ln n ) p (ln n ) 2 - 1 11. sech
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This note was uploaded on 12/09/2010 for the course MATH 1120 taught by Professor Gross during the Fall '06 term at Cornell University (Engineering School).

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1120_11 - Week 11 Outline The Integral Test p-series 1 np...

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