Write out $x_n(t)$ for the following differential equation:
$$\frac{dx}{dt} = f(t,x) = x^2, x(0)=1$$
So using the Picard Iteration : $x_n(t) = x_0 + \int_{0}^t f(s,x_{n-1}(s)) ds$,
we have $x_0(t) = 1 \quad \forall t$,
$x_1= 1+ \int_0^t (1)^2 ds = 1+t$
$x_2 = 1+ \int_0^t (1+s)^2 ds = 1+t+t^2+ \frac{t^3}{3}$
$x_3 = 1+ \int_0^t (1+s+s^2+ \frac{s^3}{3})^2 ds = 1+t+t^2+ \frac{t^3}{3}+ \frac{2}{3}t^4+ \frac{1}{3}t^5+ \frac{1}{9}t^6+ \frac{1}{56}t^7$
I am stuck in giving a general $x_n(t)$ since I could not figure out the pattern.
EDIT:
Yes I do know the solution is $\frac{1}{1-t}$, which is why I wonder there should be a recognizable pattern $1+t+t^2+t^3+...$ or something but yet my calculation seems to lead me to nowhere.