Calculus II Notes 12.3 - Calculus II-Stewart Dr. Berg...

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Calculus II- Stewart Dr. Berg Spring 2010 Page 1 12.3 12.3 The Integral Test The Integral Test Theorem A series with non-negative terms converges, if and only if, the sequence of partial sums is bounded. Proof: If the terms are non-negative, then the sequence of partial sums is non-decreasing, so the series converges if and only if the sequence of partial sums is bounded. Theorem The Integral Test If f is continuous, decreasing, and positive on [1, ) , then f ( k ) k = 1 converges if and only if f ( x ) dx 1 converges. Proof: f ( x ) dx 1 converges if and only if the sequence f ( x ) dx 1 n { } converges. Now, f (2) + f (3) +…+ f ( n ) f ( x ) dx 1 n f (1) + f (2) f ( n 1) by the definition of the integral. If f ( x ) dx 1 n { } converges, then it is bounded, so f (2) + f (3) f ( n ) { } is bounded, and since it is monotone, f ( k ) k = 1 converges. If f ( x ) dx 1 n { } diverges, then, since it is monotone, it must be unbounded, so f + f (2) f ( n { } is unbounded, so f ( k ) k = 1 must diverge.
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Calculus II Notes 12.3 - Calculus II-Stewart Dr. Berg...

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