I am playing around with using elementary techniques to derive the Taylor Series for $e^x$.
Consider the sequence of integrals $$I_n = \int_0^x t^n e^{-t} dt$$ It can be shown by induction that $$I_n = n! \left ( 1-\frac{1}{e^x}\sum_{k=0}^{n} \frac{x^k}{k!}\right)$$
I want to then consider taking $n\to \infty$ to establish the Taylor Series. This of course relies on $$\lim_{n\to\infty} \frac{I_n(x)}{n!}=0$$ which doesn't seem to easy to show for all $x$.
Any ideas would be fantastic! I am hoping that there is something elementary to use here.
To finish your proof, take the original integral and bound it like so
$$\left|\frac{I_n}{n!}\right | = \left| \int_0^x \frac{t^n}{n!}e^{-t}\:dt\right | \leq \frac{|x|^{n+1}}{n!} \to 0$$