Let $f_n(x):=n\left(e^{\frac{x^2}n}-1\right)$ on $\left[0,M\right]$. I believe this converges monotonically to $x^2$.
I used L'Hopital's rule to show pointwise convergence and the ratio of the derivatives I got converges monotonically to $x^2$ which I know implies convergence. But does it imply that the original $f_n$ also converge monotonically?