$\"\"$

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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 correspondimg integral in the terms of $\"\"$ is $\"\"$ .

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If $\"\"$ .

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$\"\"$

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$\"\"$

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The integral is divergent if $\"\"$ .

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If $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$ .

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$\"\"$

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If $\"\"$ .

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$\"\"$

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If $\"\"$ .

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$\"\"$

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Hence the integral is converges for $\"\"$ .

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Threfore, the series is converges for $\"\"$

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$\"\"$

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