# 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

Find

#### Explanation:

Consider

#(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

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

That is

That is: the radius of convergence is