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Root vs. ratio in determining radius of convergence of a power series

  1. Mar 16, 2009 #1
    Hi everyone :smile:

    When determining the radius of convergence of a power series, when should I use the ratio (a[sub n+1] / a[sub n]) test versus the root (|a[sub n]|^(1/n)) test?

    I know that I'm supposed to use the ratio only when there are factorials, but other than that, are these tests basically interchangeable?

    Also, are there any differences in usage/application of the tests in the context of determining the radius of convergence of a power series?

    Last edited: Mar 16, 2009
  2. jcsd
  3. Mar 16, 2009 #2
    So, for instance, how do I determine the radius of convergence of:


    Thanks again :smile:
    Last edited: Mar 16, 2009
  4. Mar 16, 2009 #3
    Hi brush!

    Both the ratio and the root test "work" in all cases (they are proven theorems), but which one is easier to use depends on the concrete form of the a_n. The ratio test is probably used more often then the root test.

    In your particular example, try the ratio test. :wink:
  5. Mar 16, 2009 #4
    Thank you for the reply, yyat! :smile:

    The confusion I was having was because I kept getting different results from the ratio and root test, but I have figured out what I was doing wrong.

    In the case above, both the root and ratio tests should yield (I think):

    [tex]\left|x\right| = 1[/tex]
  6. Mar 16, 2009 #5
    Yes, the radius of convergence is 1, that means the series converges for |x|<1. Note though that this tells you nothing about convergence at points with |x|=1, that needs to be checked separately.
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