Test for convergence of the series when
I am confused, since 'x' is in the denominator. Otherwise it could be treated as a general power series
Use the quotient test, You'll obtain that for the series is convergent if and divergent if . For is divergent (compare for example with ).
Fernando Revilla
Of course, the ratio test can be used here...
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The series converges when this limit is . So we need to evaluate the values of for which this ratio is , and since ...
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So the series is convergent when .
Since the ratio test fails when the limit is , you need to substitute into the original series and test the convergence of the series for that value.