This problem is slightly trickier. If you have trouble finding the general term of the series, try the following: Ask, what happens as you move from one term to the next:

Upstairs we multiply by 2 and by *x*, downstairs we multiply by 3 and replace *n* by *n*+1 (this really means we multiply the term by *n* and divide by *n*+1). Thus the absolute ratio will be given by

More formally, the general term of the series has the form

leading to the same absolute ratio:

Thus the limit of the absolute ratios is given by:

The series will converge as long as (the series will diverge when ).

Consequently, the radius of convergence equals ; the series will converge in an interval from to .

It is actually easier to find the radius of convergence when one uses the summation notation for the series. The general term is then already given!

Thu Jul 11 14:07:06 MDT 1996

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