How to find the power series and the interval of convergence for f(x)=x^2/(1+x^2)

Question

How to find the power series and the interval of convergence for f(x)=x^2/(1+x^2)

Answer 1

f(x) = x²(1 + x²) we can consider as sum of infinite geometriic series with first term a₁= x² and common ratio r = - x².

Because f(x) = x²/(1 + x²) = x² - x⁴ + x⁶ - x⁸ + ... +(-1)^{n + 1}x²ⁿ + ....

For convergence -1< r < 1 or -1 < - x² < 1, or -1 < x < 1, this is interval of convergence