$\"\"$

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Integral Test :

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Integral test: \ \

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The function $\"\"$ is a continuous, positive, decreasing function on $\"\"$ and let $\"\"$ . \ \

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(i) If $\"\"$ is convergent, then $\"\"$ is convergent. \ \

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(ii) If $\"\"$ is divergent, then $\"\"$ is divergent. $\"\"$

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The series is $\"\"$ .

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The function $\"\"$ is positive and decreasing for $\"\"$ .

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The series is converges if and only if $\"\"$ is converges.

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Consider $\"\"$ .

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Apply derivative on each side with respect to $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$ .

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Substitute $\"\"$ and $\"\"$ .

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The integral is $\"\"$ .

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Substitute $\"\"$ and $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$

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The integral is converges if and only if $\"\"$ .

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$\"\"$

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The series $\"\"$ is converges if $\"\"$ . \ \