Write a series of piecewise linear functions that converges to $f(x) = x^2$ on the interval $[0,1]$.
A sequence of piecewise linear functions that converges to $f(x) = x^2$ is as follows:
For $k \in [0,j-1]$:
\begin{align*} f_j(x) &= \begin{cases} \frac{1}{j} \cdot x & \text{if $0 \le x \le \frac{1}{j}$} \\ \frac{2k+1}{j} \cdot \left(x - \frac{k}{j} \right) + \frac{k^2}{j^2} & \text{if $\frac{k}{j} \le x \le \frac{k+1}{j}$} \\ \frac{2j-1}{j} \cdot \left(x - \frac{j-1}{j} \right) + \frac{j^2-2j+1}{j^2} & \text{if $\frac{j-1}{j} \le x \le 1$} \\ \end{cases} \\ \end{align*}
\begin{align*} \lim\limits_{j \to \infty} f_j(x) &= f(x) = x^2 \\ \end{align*}
What would be a series of piecewise linear functions such that $\sum_{j=0}^\infty f_j(x) = f(x) = x^2$?