$\"\"$

\

Consider the power series $\"\"$ such that $\"\"$ for all $\"\"$ and $\"\"$ .

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Let us assume that $\"\"$ , for every fixed $\"\"$ consider the sequence $\"\"$ .

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Consider $\"\"$ .

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$\"\"$

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Since $\"\"$ we have $\"\"$ .

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$\"\"$

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Since $\"\"$ is constant we have that $\"\"$ .

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Apply Ratio test on the series $\"\"$ .

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The ratio test states that

\

(1) If

\"\"

then the series converges absolutely.

\

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(2) If

\"\"

then the series does not converge.

\

\

(3) If

\"\"

or the limit fails to exist then the test is inclusive because there exist both convergent and divergent series that satisfy this case.

\

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Therefore the series

\"\"

is divergent when

\"\"

and is convergent when

\"\"

.

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Hence the series is convergent when

\"\"

and diverges when

\"\"

.

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$\"\"$

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If $\"\"$ then we have $\"\"$ .

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Hence we have $\"\"$ .

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If $\"\"$ and $\"\"$ if $\"\"$ .

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Therefore by the ratio test the series is divergent for every $\"\"$ which give us the radius of the power series is $\"\"$ .

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$\"\"$

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By ratio test the series $\"\"$ is divergent for every $\"\"$ where the radius of the power series is $\"\"$ .