Recursive function converges to $e^t-t-1$

Question

The recursive function $\phi_k:\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}$ is defined as: $$\phi_0(t):=0,$$ $$\phi_{k+1}(t):=\int_0^t s+\phi_k(s) ds$$

1. Prove by Induction that $\phi_k(t)=\frac{t^2}{2}+\frac{t^3}{3!}+...+\frac{t^{k+1}}{(k+1)!}$ for $k \geq1$
2. Show that the sequence of functions $(\phi_k)_{k \in \mathbb{N}}$ converges pointwise to $\phi(t)=e^t-t-1$
3. Show that $\phi$ satisfies the differential equation $$y'(t)=t+y(t), y(0)=0$$

This is how I solved 1 (I have shortened it a bit here): $$\phi_1(t)=\int_0^ts+\phi_0(s)=\int_0^ts+0=\frac{s^2}{2}$$ $$\phi_{k+1}(t)=\int_0^ts+\phi_k(s)=\int_0^ts+\frac{s^2}{2}+\frac{s^3}{3!}+...+\frac{s^{k+1}}{(k+1)!}=\frac{t^2}{2}+\frac{t^3}{3!}+...+\frac{t^{k+1}}{(k+1)!}$$ For task 3 you only have to insert the formula from task 2. My problem lies in task 2. How do I get from $e^t$ to $(\frac{t^2}{2}+\frac{t^3}{3!}+...+\frac{t^{k+1}}{(k+1)!})^t$ ? I guess I would have to use the Multinomial theorem but I don't really know how to apply it in this context.

Answer 1

First, the expression $\phi_{k}(x)$ only makes sense when $k$ is a natural number. So the correct type of $\phi$ is $\phi: \mathbb{N} \times \mathbb{R} \to \mathbb{R}$.

Once you have 1, it's easy to see that $\lim\limits_{k \to \infty} \phi_{k}(x) = \lim\limits_{k \to \infty} \sum\limits_{j = 2}^{k + 1} \frac{x^j}{j!} = \sum\limits_{j = 2}^\infty \frac{x^j}{j!}$. And we know that $e^x = \sum\limits_{j = 0}^\infty \frac{x^j}{j!} = 1 + x + \sum\limits_{j = 2}^\infty \frac{x^j}{j!}$, so $e^x - x - 1 = \sum\limits_{j = 2}^\infty \frac{x^j}{j!}$ as well.