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L18 Integral Test

# L18 Integral Test - n 2 5 p ± Series For what values of...

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Lecture 18: Integral Test Key Question: In general, it is di±cult to ²nd the exact sum of an in²nite series. We want to determine whether the series is convergent or divergent without having to ²nd the sum. General Idea: The behavior of some in²nite series is similar to the behavior of an improper integral of its related function. Compare Z 1 1 1 x and 1 X n =1 1 n 1

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The Integral Test If f is a continuous , positive and decreasing function on the interval [1 ; 1 ) and let a n = f ( n ) : Then 1) If R 1 1 f ( x ) dx is ±nite, then 1 X n =1 a n is convergent. 2) If R 1 1 f ( x ) dx is not ±nite, then 1 X n =1 a n is divergent Note: It is not necessary to begin the series at n = 1 ; nor must f be always decreasing, but rather decreasing past some number N . 2
ex. Determine whether the series converge: 1. 1 X n =1 1 n 2 + 1 3

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2. 1 X n =2 ln n n 4
3. 1 X n =2 1 n ln n 4. 1 X n =1 ne ±

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Unformatted text preview: n 2 5 p ± Series For what values of the constant p does the following p ± series converge? 1 X n =1 1 n p p ± Series Test: 8 > > > > < > > > > : 1 X n =1 1 n p is convergent when p > 1, 1 X n =1 1 n p is divergent when p ² 1 : ex. 1 X n =1 1 n : 9 ( div;p < 1) ex. 1 X n =1 1 n 4 ( con;p > 1) 6 ex. Determine the value of p so that the series converges. 1 X n =3 1 n ln n [ln(ln n )] p ( p > 1) 7 NYTI: ex. Determine whether the series converges. 1 X n =1 1 n 3 + n (conv.) ex. For what values of p would the series con-verge? a) 1 X n =1 1 n (ln n ) p b) 1 X n =1 (ln n ) p n Tonight’s Homework: Practice Integral test problems 8...
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L18 Integral Test - n 2 5 p ± Series For what values of...

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