Show that the series $\sum_{n=1}^{\infty}(e^{\frac{x}{n}}-1-\frac{x}{n})$ converges uniformly in $\left[-A,A\right]$ for any $A>0$.Show also that the sum of the series is a function with derivatives of all orders in $\mathbb{R}$.
I tried to use the Weierstrass M test, by trying $sup_{x\in \left[-A,A\right]}(e^{\frac{x}{n}}-1-\frac{x}{n})=e^{\frac{A}{n}}-1-\frac{A}{n}$, but could not prove the convergence then.