proving that the series $ 1+2^{s}+3^{s}++ $ is divergent but borel summable

Question

suppose that in the sense of distribution $ \int_{0}^{\infty}dxx^{n}T(x,s) =n^{s} $

for some distribution $ T(x) $ i do not know :(

so if we apply borel generalized resummation

$$ 1+2^{s}+3^{s}+......= \int_{0}^{\infty}( \sum_{n=1}^{\infty}\frac{x^{n}}{n^{s}})T(x,s) $$

the first term inside the integral is just $ Li_{s}(x) $

but now i am stuck because i do not know how $ T(x,s) $ should be to get a coherent result for borel resummation