Is it true that if
$\lim\limits_{x\to1^-}a_1x+a_2x^2+a_3x^3+...=C$ exists
then it is necessarily true that
$\lim\limits_{s\to0}\frac{a_1}{1^s}+\frac{a_2}{2^s}+\frac{a_3}{3^s}+... =C$
It seems like it must be true but I am not sure and don't want to make any assumptions that it is. It looks very similar to Zeta function regularization but ZFR concerns sums in the form
$\frac{1}{a_1^s}+\frac{1}{a_2^s}+\frac{1}{a_3^s}+...$ Which is quite different to my series.