The Ratio Test

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 tex2html_wrap_inline113 (the series will diverge when tex2html_wrap_inline115 ).

Consequently, the radius of convergence equals tex2html_wrap_inline117 ; the series will converge in an interval from tex2html_wrap_inline119 to tex2html_wrap_inline117 .

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!

Try it yourself!

Find the radii of convergence of the following power series: Click on the problem to see the answer, or click here to continue.

Helmut Knaust
Thu Jul 11 14:07:06 MDT 1996

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