dan picked 7 plums and 6 apples from the orchard. he gave 3 plums to sandy how many plums does he have left?
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I'm guessing the series is supposed to be
By the ratio test, the series converges if the following limit is less than 1.
The first
terms in the numerator's denominator cancel with the denominator's denominator:
also cancels out and the remaining factor of 
can be pulled out of the limit (as it doesn't depend on ).
which means the series converges everywhere (independently of
), and so the radius of convergence is infinite.
By the ratio test, the series converges if the following limit is less than 1.
The first
which means the series converges everywhere (independently of