Prove the series $\sum\limits_{n=0}^\infty\frac{x^n}{n!}$ does not converge uniformly on $\mathbb{R}$.
So what I am thinking is that the pointwise summation is $e^x$ and that I need to show there is no $n\geq N$ so that $|\sum\limits_{n=0}^\infty\frac{x^n-e^xn!}{n!}| \leq \epsilon$, for all x . We haven't covered integration; if someone could give me a hint on how to show the series does not converge uniformly, that we be great. Thanks in advance!
Best Answer
The difference between the function and the partial sum $$e^x-\sum\limits_{n=0}^k\frac{x^n}{n!}$$ is bounded? Answer: no. Why?