$\"\"$

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The function is $\"\"$ .

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$\"\"$

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

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$\"\"$

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$\"\"$ .

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The power series form of $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$ .

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$\"\"$

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$\"\"$ .

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Apply integral on each side.

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$\"\"$

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$\"\"$ .

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Find the $\"\"$ .

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$\"\"$

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Substitute $\"\"$ in eqatuion $\"\"$ .

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$\"\"$

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$\"\"$ .

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Substitute $\"\"$ in $\"\"$ .

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$\"\"$ .

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The power series of $\"\"$ is $\"\"$ .

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$\"\"$

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Find the radius of convergence.

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Consider $\"\"$ .

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Ratio test :

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Let $\"\"$ be a series with non zero terms.

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1. $\"\"$ converges absolutely if $\"\"$ .

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2. $\"\"$ diverges if $\"\"$ or $\"\"$ .

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3. The ratio test is inconclusive if $\"\"$ .

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

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Find $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$ .

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

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$\"\"$

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$\"\"$

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$\"\"$ .

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Therefore, the radius of the convergence is $\"\"$ .

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$\"\"$

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Graph :

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Graph the function $\"\"$ .

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The power series of $\"\"$ is $\"\"$ .

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Graph the partial sums are $\"\"$ , $\"\"$ , $\"\"$ and $\"\"$ .

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$\"\"$

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Observe the graph :

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As the value of $\"\"$ increases the approximation gets closer to $\"\"$ over a wider interval.

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$\"\"$

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The power series of $\"\"$ is $\"\"$ .

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Graph of the function $\"\"$ .

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Graph the partial sums are $\"\"$ , $\"\"$ , $\"\"$ and $\"\"$ .

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$\"\"$

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As the value of $\"\"$ increases the approximation gets closer to $\"\"$ over a wider interval.