Radius of Convergence

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hardyaa1

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Jul 13, 2009
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The question asks for the radius of convergence of the power series SUM(m=0, infinity) [(-1)^m / 8^m] x^3m

All our teacher told us in class was to take the limit (m -> infinity) |a_n| / |a_n+1| , and that will be the radius... i'm not sure i understand what this means,
but in the case of this problem, would it be the limit of [(-1)^m / 8^m] / [(-1)^m+1 / 8^m+1] = | 1 / 1 / -1 / 8 | = 8 ?
 
a_m = [(-1)^m / 8^m] x^(3m)

|a_(m +1)/a_m| = |x^(3m+3)/8^(m+1) ÷ x^(3m)/8^m| = |x^(3m+3)/x^(3m)| • |8^m/8^(m +1)| = |x^3|/8 < 1
|x^3| < 8
|x| < 2

The radius of convergence is 2.
 
Okay, cool. I'm starting to get it. The thing I still don't understand is how you decide what a_m is. if a power series is (SUM) a_m (x - x_0)^m then wouldn't a_m be anything without a variable in it?
 
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