So I am going to train my skills in this course. I have to work with this exercise:
"Let $ϕ_0 (t)=0$ and use the method of successive approximations to solve the given IVP. \begin{equation}y'=2(y+1),y(0)=0\end{equation}
(a) Determine $ϕ_n(t)$ for an arbitrary value of $n$.
(b) Express $\lim_{n→∞}ϕ_n(t)=ϕ(t)$ in terms of elementary functions, that is solve the given IVP. "
So I already did part (a), and obtain $ϕ_n(t)=\sum_{k=1}^n\frac{(2t)^k}{k!}$
The part (b) is annoying me. I tried to compute $\lim_{n→∞}ϕ_n(t)=ϕ(t)$ using Maple and obtain $ϕ(t)=e^{2t}-1$ my own idea is to use infinite geometric series but I am not sure how to find the function ϕ(t) using the finite geometric series.
Any help is appreciated!