renormalization - Casimir effect regularization for every divergent sum or series

can we use the tools of renormalization of casimir effect to get finite results for any divergent series in QFT?

for example let be the divergent series $\sum_{n=1}^{\infty}n^{l}$ for positive 'l' then instead of the sum we interpretate this as the difference

$$\sum_{n=1}^{\infty}n^{l}e^{-n\epsilon}-\int_{0}^{\infty}dtt^{l}e^{-t\epsilon}=finite$$

this is made to evaluate the infinite sums of the vaccuum energy in casimir effect but could it be done for any divergent series??