How do you find a power series representation for #f(x) = x / (1+x^2) # and what is the radius of convergence?

1 Answer
Sep 29, 2015

Write out a power series that when multiplied by #1+x^2# gives #x#.

Find #sum_(n=0)^oo (-1)^n x^(2n+1)# works and has radius of convergence #1#.

Explanation:

Consider #sum_(n=0)^oo (-1)^n x^(2n+1) = x - x^3 + x^5 - x^7 +...#

#(1+x^2)sum_(n=0)^oo (-1)^n x^(2n+1)#

#=sum_(n=0)^oo (-1)^n x^(2n+1) + x^2 sum_(n=0)^oo (-1)^n x^(2n+1)#

#=sum_(n=0)^oo (-1)^n x^(2n+1) - sum_(n=1)^oo (-1)^n x^(2n+1)#

#=(-1)^0x^1=x#

So:

#sum_(n=0)^oo (-1)^n x^(2n+1) = x / (1+x^2) = f(x)#

...if the sums converge.

The sum #sum_(n=0)^oo (-1)^n x^(2n+1)# is a geometric series with common ratio #-x^2#.

To converge, the absolute value of the common ratio must be less than #1#.

That is #abs(-x^2) < 1#, so #abs(x) < 1#

That is: the radius of convergence is #1#.