$\"\"$

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Direct comparison test:

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Let $\"\"$ for all $\"\"$ .

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1.If $\"\"$ convergence, then $\"\"$ convergence.

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2.If $\"\"$ diverges, then $\"\"$ diverges. $\"\"$

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The series is $\"\"$ . Here $\"\"$ and $\"\"$ .

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Observe that $\"\"$ .

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$\"\"$ .

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The series is in the form of geometric series.

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The general form of geometric series is $\"\"$ .

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Here $\"\"$ and $\"\"$ .

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$\"\"$ is geometric series.

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$\"\"$

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Convergence of a geometric series:

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A geometric series with common ratio $\"\"$ diverges if $\"\"$ .

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If $\"\"$ then the series converges to the sum $\"\"$ .

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$\"\"$ with ratio $\"\"$ .

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The series is converges to the sum of series.

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$\"\"$ .

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

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Using the direct comparision test if the series $\"\"$ is converges, then $\"\"$ is converges.

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Therefore, the series $\"\"$ is converges.

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$\"\"$

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The series $\"\"$ is converges.