Apply Weierstrass $M$-test in any bounded domain to show that the sequence is uniformly convergent on any bounded domain.
For non-convergence on any set $\mathbb{R}$, note that if the sequence converges uniformly over $\mathbb{R}$ then
\[
\lim_{n \to \infty} \sup_{x \in \mathbb{R}} \left\vert f_n(x) \right\vert = 0.
\]
Solution: Given that the series is
\[
f(x) = \sum_{n=0}^{\infty } \frac{x^n}{n}, \quad x \in \mathbb{R} .
\]
Let
\[
f_n(x) = \frac{x^n}{n!}, \quad n \in \mathbb{N} .
\]
We wil use Weierstrass $M$ test to determine if the sequence is uniformly convergent onto.
For a sequence of functions $\left( f_n \right) $ defined on a set $D$ if there exists a sequence of non-negative numbers $\left( M_n \right) $ such that
$\vert f_n(x) \vert \leq M_n \quad \forall\ n \geq 1 \text{ and } x \in D,$ and
Then the series $\sum_{n}f_n(x) $ converges absolutely and uniformly on $D$.
We claim that the sequence $\left( f_n \right) $ satisfies the Weierstrass condition on every bounded set $D$. Let $\alpha $ be an upper bound of the set $D$. Then
\begin{align*}
\vert f_n(x) \vert & = \left\vert \frac{x^n}{n!} \right\vert \leq \frac{\alpha ^n}{n!}.
\end{align*}
Take the sequence
\[
M_n = \frac{\alpha ^n}{n!}.
\]
Since the sequence $\left( M_n \right) $ is convergent, by Weierstrass $M$-test we conclude that $f$ converges uniformly on any bounded set.
We now claim that the given sequence does not converge uniformly on all of $\mathbb{R} $. Note that if the sequence converges uniformly then
\[
\lim_{n \to \infty} \sup_{x \in \mathbb{R}} \left\vert f_n(x) \right\vert = 0.
\]
However, observe that
\[
\sup_{x \in \mathbb{R}} \left\vert f_n(x) \right\vert = \sup_{x \in \mathbb{R}} \left\vert \frac{x^n}{n!} \right\vert = \infty,
\]
since for any fixed $n$, as $x \to \infty$, $\frac{x^n}{n!} \to \infty$. Therefore, the sequence $\left( f_n \right) $ does not converge uniformly on $\mathbb{R} $. Thus, the series $f(x) = \sum_{n=0}^{\infty} \frac{x^n}{n!}$ converges uniformly on any bounded interval but not on the entire real line.