We have the IVP
$$\dot{y}(t) = Ay(t)$$ $$y(0)=c$$
We consider the Picard-Lindelof Iteration. One step of this iteration is given by $$y_{k+1} = y_0(t) + \int_{0}^{t} A y_k (\tau) d\tau$$ where $y_0(t) = c$. We're asked to calculate the first few steps of this iteration to then determine its exact solution.
The exact solution of this IVP is given by $$y(t) = e^{At}y_0$$
Recalling the Taylor Expansion of the exponential function, we suppose $$y_k(t) = (\sum_{j=0}^{k} \frac{(At)^j}{j!})y_0$$
So we obtain $$y_{k+1}(t) = y_0 + \int_{0}^{t} A y_k (s) ds = y_0 + (\sum_{j=0}^{k} \frac{A^{j+1}}{j!}\int_{0}^{t} s^j ds)y_0 = y_0 + (\sum_{j=0}^{k} \frac{(At)^{j+1}}{(j+1)!})y_0 = (\sum_{j=0}^{k+1} \frac{(At)^{j+1}}{j!})y_0$$
Thus $$\lim_{k \to \infty} y_{k+1}(t) = e^{At}y_0 = y(t)$$