Let
$$ f_{n}: \mathbb{R} \xrightarrow{} \mathbb{R}, x \mapsto f_{n}(x) = \left( 1 + \frac{x}{n}\right)^{n} $$
Now we know that for all $x \in \mathbb{R}$ we have
$$ \lim_\limits{n \to \infty} f_{n}(x) = \lim_\limits{n \to \infty} \left( 1 + \frac{x}{n}\right)^{n} = \exp(x) $$ pointwise. My question is now, whether the sequence $\{f_{n}\}_{n \in \mathbb{N}}$ converges uniformly to $\exp(x)$
That $f_{n}$ converges uniformly on any compact interval follows immediatley from Dini's Theorem. Does it also converges uniformly on any finite subinterval of $\mathbb{R}$?
On the whole real line, no. That’s because for every $n$ you can choose $x$ accordingly so that the difference btw exponential and $(1+x/n)^n$ (basically a polynomial of degree $n$) becomes arbitrarely large.