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Topic: Power series problem
Replies: 2   Last Post: Jan 22, 2012 12:39 AM

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MTBrenneman@gmail.com

Posts: 983
Registered: 8/21/06
Power series problem
Posted: Jan 21, 2012 3:47 PM
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Hi,

I'm working through a power series problem from calc 3, and came
across something that I can't explain.

I want to find the Taylor series expansion and radius of convergence
for f(x) = x /[ 1 + 2x^2]

So I factor x out of the expression and work on getting power series
of g(x) = 1/(1+2x^2)

The one way to do this is to write it as [ 1 + 2x^2 ] ^(-1) = [1 -
(-2x^2) ]^(-1)
and then write out the geometric series whose terms are powers of r =
(-2x^2). The condition for the convergence of a geometric series |
2x^2| < 1, let's me compute the radius of convergence as 1/sqrt(2).
Taylor series is then x*Power series for g(x)

BUT, when I do this problem a different way, I get a different
solution:
g(x) = 1/ [1+2x^2] = -1/[-1-2x^2] = -1/[1-2(1+x^2)]
but there is no value of x which satisfies the necessary condition to
write this as a geometric series, i.e. 2(1+x^2) < 1 is satisfied for
no value of x.

Not quite sure why I cannot get the power series to work out this way.

TIA,
Matt



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