How can I prove the uniform convergence of the series: $\sum_{n \geq 0} (-1)^ne^{-a_nx}$ with $x \ \in \mathbb{R}$
$(a_n)$ is an increasing sequence of positive numbers.
I could not apply the M test because I will always get an inequlity in terms of $x$.
How can I prove that the uniform convergence is only on an intervall $]a, + \infty[$.
First of all, your series only makes sense for $x>0$. Let $I_\alpha=[\alpha, \infty)$ for $\alpha>0$. Then your series converges uniformly for $x\in I_\alpha$ as long as $\sum_{k\ge 0} e^{- a_n\alpha}$ converges. Indeed, by the $M$-Test, you have that $e^{-xa_n}\le e^{- a_n\alpha}\,\forall n\ge 0$ with the latter being summable. However, your series cannot converge uniformly on $(0,\infty)$ because $\sup_{x\in(0,\infty)}e^{-x a_n}=1$ for all $n$. Indeed, if the series did converge uniformly, that limit must be precisely zero.