The Ratio Test

Let's start by writing out the absolute ratios:


As in the previous example, we can simplify tex2html_wrap_inline86 .

For the first term we want to use the hint, so let's rewrite


Do you feel like visiting our Algebra section? Just kidding...

Having jumped this hurdle, we can compute the limit of the ratios as


Thus the radius of convergence is tex2html_wrap_inline88 , the series will converge in the interval from tex2html_wrap_inline90 to tex2html_wrap_inline92 .

Seeing is believing? Here are the pictures of the first terms of the power series:

Click here for the animated GIF file.

Draw the graph of the power series on its interval of convergence. The behavior at the left is different from the behavior at the right. Try to explain the different behaviors!

Where to go next?

Go to the Taylor series pages.

Helmut Knaust
Fri Jul 12 16:21:45 MDT 1996
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