Non-trivial examples of power series which are uniformly convergent on $[0,1)$ and left-continuous at $x = 1$

Question

The question is motivated by a more extensive problem that needs a formal proof, but I am not interested in help on the proof itself, but I'd like to see some examples of such power series.

I put non-trivial in the title because it has turned out that power series such as:

$$e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} $$

...are not very interesting as they are uniformly convergent on $\mathbb{R}$ and "trivially" left-continuous at $x = 1$ due to regular continuity at $x = 1$. The power series I have toyed with do not offer insight or intuition on the proof I am working on, so I am looking for some example that cannot be trivially deduced to be left-continuous at $x = 1$.

Summary : I am looking for a non-trivial power series such that $\sum_{n=0}^{\infty} a_n x^n$ converges uniformly on $[0,1)$ and has the property that for its limit function $f(x)$ we have: $$ \lim_{x \to 1^-} f(x) = f(1)$$

Thank you!

Answer 1

Using Abel's Theorem you can come up with lots of examples, for example

$$\sum_{n=1}^\infty\frac{x^n}{n^p}\;,\;\;\forall\,p>1\;,\;\;\text{with convergence radius}\;\;R=1$$

Answer 2

Why not just take half a circle? Over $[-1,1]$, we have: $$1-\sqrt{1-x^2} = \sum_{n\geq 0}\frac{x^{2n+2}}{(2n+2)\,4^n}\binom{2n}{n}.$$