For $x$ in $\mathbb{R}$, let $f(x)=(x+1) e^{x}$. After finding the $k$ th Taylor polynomial for $f$ at $x_{0}=0$ and the corresponding Lagrange form of the remainder $R_{k}(0 ; x)$,
prove that $$\lim _{k \rightarrow \infty} R_{k}(0 ; x)=0$$ (uniformly) on $\mathbb{R}$.
I have found the $k$ th derivative as $(x+k+1)e^x$, and $R_{k}(0 ; x)= \frac{(c+k+2)x^{k+2}}{(k+1)!} e^c$ for some $c$ lying strictly between $0$ and $x$. However, since the domain is considered as $\mathbb{R}$, I don't know how to proceed for proving the uniform convergence of the remainder using the limit. Are there any alternative approaches for it, such as dividing the domain into finite subintervals or maybe considering $x$ as the fixed value?