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# 12-3 - 12.3 The Integral Test The formula n 2n used to dene...

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12.3: The Integral Test The formula n 2 n used to define the terms in X n =1 n 2 n also defines a function on the real line f ( x ) = x 2 x so that a n = f ( n ) for all n . The Integral Test compares the area under the graph of f ( x ) with the area represented by n =1 n 2 n . Theorem (The Integral Test) . Suppose f is a continu- ous, positive, decreasing function on [1 , ) and f ( n ) = a n . Then the series n =1 a n is convergent if and only if R 1 f ( x ) dx is convergent. In other words, 1. If R 1 f ( x ) dx is convergent, then so is n =1 a n . 2. If R 1 f ( x ) dx is divergent, then so is n =1 a n . 1

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12.3: Examples Determine whether the series is convergent or divergent. 1. n =1 1 (2 n +1) 5 2. n =1 n 2 n 3 +1 3. n =1 ln( n ) n 3 2
12.3: The p -Series For a constant p , the p -Series is n =1 1 n p . Much like the Geometric Series, we will find that some series are just a p -Series in disguise. We can use the Integral Test and the Divergence Test to determine when n =1 1 n p is convergent or divergent.

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12-3 - 12.3 The Integral Test The formula n 2n used to dene...

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