Taylor Series for 1/(1+x^2) without Substitution

[sparkle123]

Homework Statement

How do we get that the Taylor Series of 1/(1+x^2) around x= 0
is 1 - x^2 + x^4 + ... + (-1)^n x^{2n} + ... for |x|<1, without using a substitution of x=-x^2 into the Taylor series for 1/(1-x)?

Homework Equations

The Attempt at a Solution

 

[vela]

You could go back to the definition of the Taylor series. Calculate the derivatives, etc.