Another "not-yet-answer"...
I've tried another idea. Assume the function f(x) is expressed by the following composition: $$\small x' = \exp(x)-1 $$ $$\small f(x) = g(x') = g(exp(x)-1) $$ The idea is, that the unavoidable big "hump" in the partial sums, after which the sequence of partial sums begins to decrease, may be absorbed by the function $\small g(x)$ - because $\small \exp(x) $ is really small for large negative x and x' is then very little above -1. I did not yet arrive at a conclusive result; but the power series for $\small g(x) $ begins with the smooth looking form (and gives the partial sums for $\small x'=\exp(-100)-1 $):
$\qquad \small \begin{array} {r|r} \text{powerseries} & \text{partial sums for x' } \\ \hline \\ 1.00000000000 & 1.00000000000 \\ +1.00000000000x & 3.72007597602E-44 \\ +0.207106781187x^{2} & 0.207106781187 \\ +0.0344748426106x^{3} & 0.172631938576 \\ -0.0100670743762x^{4} & 0.162564864200 \\ +0.00821765977664x^{5} & 0.154347204423 \\ -0.00654357122833x^{6} & 0.147803633195 \\ +0.00537330847179x^{7} & 0.142430324723 \\ -0.00451702185603x^{8} & 0.137913302867 \\ +0.00386915976824x^{9} & 0.134044143099 \\ -0.00336528035075x^{10} & 0.130678862748 \\ +0.00296428202807x^{11} & 0.127714580720 \\ -0.00263893325448x^{12} & 0.125075647465 \\ +0.00237058888853x^{13} & 0.122705058577 \\ -0.00214611388717x^{14} & 0.120558944690 \\ +0.00195602261228x^{15} & 0.118602922077 \\ -0.00179331457091x^{16} & 0.116809607506 \\ +0.00165272361723x^{17} & 0.115156883889 \\ -0.00153022060566x^{18} & 0.113626663284 \\ +0.00142267593977x^{19} & 0.112203987344 \\ -0.00132762563657x^{20} & 0.110876361707 \\ +0.00124310598493x^{21} & 0.109633255722 \\ -0.00116753462507x^{22} & 0.108465721097 \\ +0.00109962364925x^{23} & 0.107366097448 \\ \end{array} $
The the question is, for some large negative x, say $\small x=-100 \qquad x'=exp(-100)-1 = -1+ \epsilon $ the series $\ g(x') $ converges to zero. Unfortunately - although we've translated the original problem to one with nice small numbers I don't see, how to really come nearer a solution, because the convergence of $\small g(-1+\epsilon) $ is really slow - if it converges at all to a positive value... So this is not yet a solution, but perhaps a suggestion for a path to try...
