Question

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

Answer 1

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`.