dy/dt=y^2 with initial condition y(0)=1

Prove that every picards iteration is defined for every t, but the exact solution is defiuned for t<1. Show that Taylor series for the exact solution converges for |t|<1.

The picards iterations are all polynomials so are all defined for every t.

The solution I have is $\displaystyle y = (1 - t)^{-1}$. Is defined for every t except 1?. And I couldnt find taylor series.
Thanks for the answers.