A problem with sums 1+2+3+4+5+6

[zetafunction]

using an exponential regulator $$exp(-\epsilon n)$$ the sum

$$1+2+3+4+5+6+7+...= -1/12+ 1/\epsilon^{2}$$

and for Casimir effect $$1+8+27+64+125+...= -1/120+ 1/\epsilon^{4}$$

can i simply remove in the calculations of divergent series 1+2+3+4+5.. and similar the epsilon terms imposing renormalization conditions ??

how about for the rest of sums $$1+2^{m}+3^{m}+...= \zeta (-m) + 1/\epsilon ^{m+1}$$

if i introducte a power regulator $$n^{-s}$$ in the limit s-->0+ i would get

$$\zeta(s-m)=\zeta(-m)$$ but i am not sure, why this work

for example in the definition of a functional determinant (in differential geommetry )

$$\prod_{i} \lambda_{i}$$ apparently there is no divergent term proportional to $$log(\epsilon)$$ as one would expect since the product is divergent

 

[arivero]

Now I see the series, a related problem comes from Conway via Baez:
Let $$1+2^{m}+3^{m}+...+ N^m= U^m$$
For which values of N, m do we get an integer value for U?

In some strange way, string theory relates the solution N=24 (or 26?), m=2 with the regulation of N=infinity, m=1