Problem: Given a sequence $(f_n)_{n\in\mathbb{N}}$ of polynomial functions such that $f_n(x)=\sum_{j=0}^{n}\dfrac{j^3}{4^{j+2}}x^j$, find the largest real number $r$, such that $(f_n)_{n\in\mathbb{N}}$ converges to a function $f:(0,r)\to\mathbb{R}$ pointwise on $(0,r)$. Also show that $f$ is continuous on $(0,r)$.
Question: I was able to show pointwise convergence on the maximal interval $(0,4)$ by treating $(f_n(x))_{n\in\mathbb{N}}$ as a power series and calculating its radius of convergence. However, I haven't been able to prove that $f$ is continuous on $(0,4)$, as the only thing that came to my mind was to use that polynomial functions are continuous.
Let $x\in (0,4)$. Choose $\alpha>0$ with $x+\alpha<4$. Then, for any $y\in (0,4)$ and $m>n$, \begin{align} |f_m(y)-f_n(y)| &\leq \sum_{k=n+1}^m \frac{k^3}{4^{k+2}}\,|y|^k\\ \ \\ &\leq \sum_{k=n+1}^m \frac{k^3}{4^{k+2}}\,(x+\alpha)^k\\ \ \\ &=\frac1{16} \sum_{k=n+1}^m {k^3}\,\left(\frac{x+\alpha}4\right)^k\\ \ \\ &=\frac1{16} \sum_{k=n+1}^m {k^3}\,q^k\\ \ \\ \end{align} where $q=\tfrac{x+\alpha}4\in(0,1)$. As this last sum is the tail of a convergent series, we get that $\{f_n\}$ is uniformly Cauchy, so its limit is a continuous function on $(0,4)$.