$\"\"$

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The integral Test :

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If $\"\"$ is positive, continous, and decreasing for $\"\"$ and $\"\"$ then $\"\"$ and $\"\"$ either converge or both diverge.

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$\"\"$

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The integral series is $\"\"$

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The summation notation of series is $\"\"$ .

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Let the function be $\"\"$ .

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Find the derivative of the function.

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$\"\"$ .

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$\"\"$ .

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$\"\"$ the function is decreasing for $\"\"$ .

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$\"\"$ is positive, continuous and decreasing for $\"\"$ .

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$\"\"$ satisfies the conditions of Integral Test.

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Integral Test is applicable for the series series.

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$\"\"$

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$\"\"$

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Apply $\"\"$ -substitution :

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

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$\"\"$

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Substitute back $\"\"$ .

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$\"\"$ .

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$\"\"$

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The series diverges.

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The series diverges.