Power series convergent but not uniformly convergent in $\mathbb{R}$

Question

Suppose its radius of convergence $R=\infty$, I wonder if there is an example satisfying the condition in my title.

I already know when $R<\infty$ there is an example such as $\sum x^n$ in $(-1,1)$.

Any help will be appreciated.

Answer 1

No power series with infinitely many nonzero coefficients converges uniformly on all of $\mathbb{R}$. Indeed, let $f_k(x)=\sum_{n=0}^k a_nx^n$ and suppose $f_k$ converges uniformly to a function $f$ on $\mathbb{R}$. Choose any $\epsilon>0$ and choose $N$ such that $|f(x)-f_k(x)|<\epsilon$ for any $x$ and any $k\geq N$. Suppose $k> N$ is such that $a_k\neq 0$. Then we can choose $x$ large enough so that $|a_kx^k|>2\epsilon$. For such an $x$, $|f_{k}(x)-f_{k-1}(x)|>2\epsilon$, which is a contradiction since $f_{k}(x)$ and $f_{k-1}(x)$ are both within $\epsilon$ of $f(x)$.

This contradiction means that $a_k=0$ for all $k>N$. In particular, only finitely many of the coefficients $a_k$ are nonzero.