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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?

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    Davide's answer is great. Another easy way is to substitute a Taylor series for $e^{x^2/n}$.2011-11-17
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    @ Davide Giraudo I treated x as fixed and applied L'hopitals rule to the function of a real variable associated with the sequence [writing n as 1/(1/n)]2011-11-18

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