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 <br /> 1+2^{m}+3^{m}+...+ N^m= U^m<br />
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