Does there exist a sequence of strictly increasing analytic positive functions $a_i : ]-1,1 [\to \mathbb{R}^{>0}$ such that $$f (x) = \sum_{i=0}^{+\infty} a_i (x) $$ converges for $x\leq 0$ and diverges for $x>0$?
I was able to prove the existence of such functions if we replace analytic by $\mathcal C^{\infty}$ but I am stuck here.
Thank you very much in advance for your help.
Take $a_k(x) = \frac{e^{kx}}{k^2}$. Then $$ \sum_{k=1}^\infty \frac{e^{kx}}{k^2} $$ converges for $x\leq 0$ and diverges for $x>0$.