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sec11_3

# sec11_3 - ∞ ± n =1 1 n 5 2 Estimating the sum of a...

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11.3 The Integral Test and Estimates of Sums 1. The Integral Test Theorem 1.1 (The Integral Test) . Suppose f is a continuous, positive, decreasing function on [1 , ] and let a n = f ( n ) . Then the series n =1 a n is convergent if and only if the improper integral 1 f ( x ) dx is convergent. In other words: (i) If 1 f ( x ) dx is convergent, then n =1 a n is convergent. (ii) If 1 f ( x ) dx is divergent, then n =1 a n is divergent. Example 1.1. (problem 6) Use the integral test to determine whether the series is convergent or divergent. n =1 1 n + 4 Example 1.2. (problem 21) Use the integral test to determine whether the series is convergent or divergent. n =2 1 n ln n 1

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Section 11.3 Integral Test and Estimates of Sums 2 Theorem 1.2. The p - series n =1 1 n p is convergent if p > 1 and divergent if p 1 . Example 1.3. Determine whether the series is convergent or divergent. (problem 4)
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Unformatted text preview: ∞ ± n =1 1 n 5 2. Estimating the sum of a series Theorem 2.1. Suppose f ( k ) = a k , where f is a continuous, positive, decreasing function for x ≥ n and ∑ a n is convergent. If R n = s-s n , then ² ∞ n +1 f ( x ) dx ≤ R n ≤ ² ∞ n f ( x ) dx and s n + ² ∞ n +1 f ( x ) dx ≤ s ≤ s n + ² ∞ n f ( x ) dx Example 2.1. (problem 32) (a) Find the partial sum s 10 of the series ∞ ± n =1 1 /n 4 . Es-timate the error in using s 10 as an approximation to the sum of the series. (b) Use the above inequalities with n = 10 to give an improved estimate of the sum. (c) Find a value of n so that s n is within 0.00001 of the sum....
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sec11_3 - ∞ ± n =1 1 n 5 2 Estimating the sum of a...

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